*h%XSSKJr SSKJr SSKrSSKrSSKrSSKrSSKJ r SSK J r J r J r J r JrJr SSKJrJr SSKJr SS KJrJr SS KJr SS KJrJr SS KJr SS KJ r J!r!J"r"J#r#J$r$ SSK%J&r& SSK'J(r( SSK)J*r* SSK+J,r,J-r-J.r. SSK/J0r0 SSK1r1SSK2J3r4 SSK2J5r5J6r7 SSK8J9r9 SSK2J:r:J;r;JJ?r? SSK@JArBJCrDJErFJGrGJHrIJJrJJKrK SSKLJMrM SSKNJOrO SSKPJQrQ \R"S5rSSdSjrTSrUSS0rVSeS jrWS!rXS"rY\ZR\\RS#55r^S$r_"S%S&\5r`"S'S(\`5ra\a=\ \b'\ \R'\ard"S)S*\`5re"S+S,\e5rf\f\ \g'"S-S.\5rh"S/S0\e5ri"S1S2\f5rj"S3S4\j\S59rk"S6S7\j\S59rl"S8S9\j\S59rm"S:S;\i\S59rn"S<S=\`\S59ro\Rrp"S>S?\`\S59rq"S@SA\`\S59rr\Rrs\0"\r\5SB5rt"SCSD\\S59ru\Rrv"SESF\5rw"SGSH\w\S59rx\Rry"SISJ\w\S59rz\Rr{"SKSL\w\S59r|"SMSN\w\S59r}"SOSP\w\S59r~"SQSR\w\S59r"SSST\\S59r\GRrSUrSVrSfSWjr\0"\ \`5SX5rtSYr\\ \GR '\-b?SZrS[r\\ \"\-GR"S55'\\ \"\-GR"SS55'\.b?S\rS]r\\ \"\.GR"S55'\\ \"\.GR"SS55'S^r\\ \?'S_r\\ \'SS`KJr SSaKJr \l"5\lSSbKJr \k"5\lScr\"5 \R\R\R\R4rg)g) annotations)overloadN)Tuple) SympifyError_sympy_convertersympify_convert_numpy_types_sympify_is_numpy_instance)S Singleton)Basic)Expr AtomicExpr) pure_complex)cacheit clear_cache) _sympifyit) num_digitsigcdilcm mod_inverseinteger_nthroot) fuzzy_not) NumberKind)ordered) SYMPY_INTSgmpyflint)dispatch)bitcount round_nearestMPZ)mpf_powmpf_pimpf_e phi_fixed) mpnumeric)finffninffnanfzero _normalize prec_to_dps dps_to_prec)debug)sympy_deprecation_warning)global_parametersc [U[5(a([USS9(d [S5e[U5U:H$U(dXpU(dgU(GdXpCUS:Xa[U5[U5:H$UGc[ U5[ U5pC[ SX4455(d [S5eUR [5nUR [5nU(d U(dX4:H$[S5HWn[USS9nU(aMU(a6UR[[SU5555n[USS9n OXepeXCpCMY [U5n U (d;U(a4UR[[S U5555n[USS9n W(a6U (a/US (d U S (a[ S [X555$S [[UR[US UR555-n[[!X4- 5U-5S:*$[!X- 5n [!U5n U(a U S :aX- U:*$X:*$)aReturn a bool indicating whether the error between z1 and z2 is $\le$ ``tol``. Examples ======== If ``tol`` is ``None`` then ``True`` will be returned if :math:`|z1 - z2|\times 10^p \le 5` where $p$ is minimum value of the decimal precision of each value. >>> from sympy import comp, pi >>> pi4 = pi.n(4); pi4 3.142 >>> comp(_, 3.142) True >>> comp(pi4, 3.141) False >>> comp(pi4, 3.143) False A comparison of strings will be made if ``z1`` is a Number and ``z2`` is a string or ``tol`` is ''. >>> comp(pi4, 3.1415) True >>> comp(pi4, 3.1415, '') False When ``tol`` is provided and $z2$ is non-zero and :math:`|z1| > 1` the error is normalized by :math:`|z1|`: >>> abs(pi4 - 3.14)/pi4 0.000509791731426756 >>> comp(pi4, 3.14, .001) # difference less than 0.1% True >>> comp(pi4, 3.14, .0005) # difference less than 0.1% False When :math:`|z1| \le 1` the absolute error is used: >>> 1/pi4 0.3183 >>> abs(1/pi4 - 0.3183)/(1/pi4) 3.07371499106316e-5 >>> abs(1/pi4 - 0.3183) 9.78393554684764e-6 >>> comp(1/pi4, 0.3183, 1e-5) True To see if the absolute error between ``z1`` and ``z2`` is less than or equal to ``tol``, call this as ``comp(z1 - z2, 0, tol)`` or ``comp(z1 - z2, tol=tol)``: >>> abs(pi4 - 3.14) 0.00160156249999988 >>> comp(pi4 - 3.14, 0, .002) True >>> comp(pi4 - 3.14, 0, .001) False T)or_realz$when z2 is a str z1 must be a Numberc38# UHoRv M g7fN) is_number.0is O/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/core/numbers.py comp..us3Fq{{Fzexpecting 2 numbersr5c38# UHoRv M g7fr:_precr<s r?r@rAs/DArBc38# UHoRv M g7fr:rDr<s r?r@rAs'<ArBrc3<# UHup[X5v M g7fr:)comp)r=r>js r?r@rAs>+$!4::+s rE) isinstancestrr ValueErrorr allatomsFloatrangenr0minziprEgetattrintabs) z1z2tolabfafb_cacbdiffaz1s r?rHrH*sz"cB-CD D2w"} B  1 "9q6SV# # ;1:wqzq3QF333 !677BBbv 1X!!T2rCC C/D/D,D EF)!T:!#B 1aB"CC C'<'<$<=>!!T2bber!u>#b+>>>k#aggwq'177/K"LMMCs15z#~&!+ + rwrec:UR5(d[SU-5eUR5upn[U5nUS:a[ [ U55nXT4$SU-n[ S[[U5555n[X-SU*-5nXT4$)z3Convert an ordinary decimal instance to a Rational.zdec must be finite, got %s.rrxc36# UHupUSU--v M g7f)rJN)r=r>dis r?r@,_decimal_to_Rational_prec..s=&frSgs r?_literal_floatrs( GGCLE 5zA~ 5zQ <<d $ $!"A#1 GGCLE 5zA~ Uq1#1 QQqTT\ abE  SAAYA D)) rec^\rSrSrSrSrSrSrSrSr \ r Sr Sr SrS rS rS rS rS rSrSrSrSrSrSrSrSr\S5r\S0Sj5rS1Sjr \!S2Sj5r"\!S3Sj5r"\#"S\$5S4Sj5r"\#"S\$5S5r%\#"S\$5S5r&\#"S\$5S5r'S r(S!r)S"r*S#r+S$r,S%r-U4S&jr.S'r/SS(.S)jr0S*r1S5S+jr2S5S,jr3S-r4S.r5S/r6Sr7U=r8$)6NumberiaRepresents atomic numbers in SymPy. Explanation =========== Floating point numbers are represented by the Float class. Rational numbers (of any size) are represented by the Rational class. Integer numbers (of any size) are represented by the Integer class. Float and Rational are subclasses of Number; Integer is a subclass of Rational. For example, ``2/3`` is represented as ``Rational(2, 3)`` which is a different object from the floating point number obtained with Python division ``2/3``. Even for numbers that are exactly represented in binary, there is a difference between how two forms, such as ``Rational(1, 2)`` and ``Float(0.5)``, are used in SymPy. The rational form is to be preferred in symbolic computations. Other kinds of numbers, such as algebraic numbers ``sqrt(2)`` or complex numbers ``3 + 4*I``, are not instances of Number class as they are not atomic. See Also ======== Float, Integer, Rational Trrxc[U5S:XaUSn[U[5(aU$[U[5(a [ U5$[U[ 5(a[U5S:Xa[ U6$[U[[R[R45(a [U5$[U[5(aUR5nUS:Xa[R $US:Xa[R"$US:Xa[R"$US:Xa[R$$['U5n[U[5(aU$[)SU-5eS n[+U[-U5R.-5e) Nrrr5naninf+inf-infz$String "%s" does not denote a Numberz ! cE6::w? @ @:  c3  99;Du}uu zz!zz!)))#,C#v&&  !G#!MNNLd3i00011rec,[UR5$r:)boolis_extended_negativeselfs r?could_extract_minus_signNumber.could_extract_minus_sign^sD--..rec`SSKJn [USS5(a [X5$U"X/UQ70UD6$)Nr)invertr;T)sympy.polys.polytoolsrrVr)rothergensargsrs r?r Number.invertas40 5+t , ,t+ +d1D1D11recPSSKJn [U5nUR(d[R X4;a [R [R 4$U(d [S5eUR(a7UR(a&[[URUR56$[U[5(aU[U5- nOX- nUR (arUS:a [#U5O [#U5S- nXU-- nU[U5:XaUS- nSn[U[5(d[U[5(a [U5nO(U(aU"U5U"U5:XaSOSnU(aUOUn[XE5$![ a [s$f=f)Nr)rlzmodulo by zerorrx)$sympy.functions.elementary.complexesrlr is_infiniter rrNotImplementedZeroDivisionError is_IntegerrdivmodrzrLrQrrrW)rrrlratwrs r? __divmod__Number.__divmod__gsF= "5ME155TM#9quu~%$:#$45 5 ??u//&12 2 u % %x&C*C ??1HC#c(Q,AQwAE%L Q$&&*UE*B*B!H$t*U ";"AAQ{+ "! ! "sAFF%$F%c\[U5n[X5$![a [s$f=fr:)rrrrrrs r? __rdivmod__Number.__rdivmod__s4 "5MEe"" "! ! "s ++cF[SURR-5e)z7Evaluation of mpf tuple accurate to at least prec bits.z%s needs ._as_mpf_val() methodNotImplementedError __class__rrrks r? _as_mpf_valNumber._as_mpf_vals$!"B ^^ $ $#&' 'recL[RURU5U5$r:rQ_newrrs r? _eval_evalfNumber._eval_evalfzz$**40$77recR[XR5nURU5U4$r:)maxrErrs r? _as_mpf_opNumber._as_mpf_ops&4$%t++recL[R"URS55$)N5)mlibto_floatrrs r? __float__Number.__float__s}}T--b122recF[SURR-5e)Nz%s needs .floor() methodrrs r?floor Number.floors$!"< ^^ $ $#&' 'recF[SURR-5e)Nz%s needs .ceiling() methodrrs r?ceilingNumber.ceilings$!"> ^^ $ $#&' 'rec"UR5$r:rrs r? __floor__Number.__floor__zz|rec"UR5$r:rrs r?__ceil__Number.__ceil__||~recU$r:rrs r?_eval_conjugateNumber._eval_conjugate rec<SSKJn U"[R/UQ76$)Nr)Order)sympy.series.orderrr One)rsymbolsrs r? _eval_orderNumber._eval_orders,QUU%W%%recX*:XaU*$U$r:rroldnews r? _eval_subsNumber._eval_subss %<4K recg)N)rrrrrs r? class_keyNumber.class_keysrec*UR5SSU4$)N)rrr)r)rorders r?sort_keyNumber.sort_keys~~"d22rec[er:rrs r?__neg__Number.__neg__s!!rercgr:rrs r?__add__Number.__add__s>Arecgr:rrs r?rrs,/recT[U[5(a~[R(aiU[R La[R $U[R La[R $U[RLa[R$[R"X5$r:) rLrr4evaluater rrrrrrs r?rrsm eV $ $):)C)C~uu !**$zz!!,,,)))!!$..recT[U[5(a~[R(aiU[R La[R $U[R La[R$U[RLa[R $[R"X5$r:) rLrr4rr rrrr__sub__rs r?rNumber.__sub__sm eV $ $):)C)C~uu !**$)))!,,,zz!!!$..rec[U[5(Ga[R(aU[R La[R $U[R LaRUR(a[R $UR(a[R $[R$U[RLaRUR(a[R $UR(a[R$[R $O[U[5(a[$[R"X5$r:)rLrr4rr rris_zero is_positiverrrr__mul__rs r?r$Number.__mul__s eV $ $):)C)C~uu !**$<<55L%%::%---!,,,<<55L%%---::% -u % %! !!!$..rec0[U[5(al[R(aWU[R La[R $U[R [R4;a[R$[R"X5$r:) rLrr4rr rrrZeror __truediv__rs r?r(Number.__truediv__s] eV $ $):)C)C~uu 1::q'9'9::vv %%d22recF[SURR-5e)Nz%s needs .__eq__() methodrrs r?__eq__ Number.__eq__$!"= ^^ $ $#&' 'recF[SURR-5e)Nz%s needs .__ne__() methodrrs r?__ne__ Number.__ne__r-rec[U5n[SURR -5e![a [SU<SU<35ef=f)NInvalid comparison z < z%s needs .__lt__() methodr rrrrrrs r?__lt__ Number.__lt__sX JUOE""= ^^ $ $#&' ' JD%HI I J /A c[U5n[SURR -5e![a [SU<SU<35ef=f)Nr2z <= z%s needs .__le__() methodr3rs r?__le__ Number.__le__ sX KUOE""= ^^ $ $#&' ' KT5IJ J Kr6c[U5n[U5RU5$![a [SU<SU<35ef=f)Nr2z > )r rrr4rs r?__gt__ Number.__gt__sK JUOE%%d++ JD%HI I J 'Ac[U5n[U5RU5$![a [SU<SU<35ef=f)Nr2z >= )r rrr8rs r?__ge__ Number.__ge__sK KUOE%%d++ KT5IJ J Kr=c >[TU]5$r:super__hash__rrs r?rDNumber.__hash__"w!!recg)NTr)rwrtflagss r? is_constantNumber.is_constant%rerationalcUR(dU(dUS4$UR(a[RU*44$[RU44$Nr) is_Rational is_negativer NegativeOner)rrOdepskwargss r? as_coeff_mulNumber.as_coeff_mul(sB   88O   ==D5(* *uutg~recRUR(aUS4$[RU44$rQ)rRr r')rrUs r? as_coeff_addNumber.as_coeff_add0s$   8OvvwrecXU(dU[R4$[RU4$z1Efficiently extract the coefficient of a product.r rrrOs r? as_coeff_MulNumber.as_coeff_Mul6s!; uud{recXU(dU[R4$[RU4$z3Efficiently extract the coefficient of a summation.r r'r_s r? as_coeff_AddNumber.as_coeff_Add<s!< vvt|recSSKJn U"X5$)z#Compute GCD of `self` and `other`. r)gcd)rrh)rrrhs r?rh Number.gcdB-4recSSKJn U"X5$)z#Compute LCM of `self` and `other`. r)lcm)rrl)rrrls r?rl Number.lcmGrjrecSSKJn U"X5$)z1Compute GCD and cofactors of `self` and `other`. r) cofactors)rro)rrros r?roNumber.cofactorsLs3%%rer:)returnr)rzNumber | int | floatrqr)rrrqr)rqrF)9r __module__ __qualname____firstlineno____doc__is_commutativer; is_Number __slots__rErkindrrrrrrrrrrrrrrrr  classmethodrrrrrrrrrr$r(r+r/r4r8r;r?rDrKrWrZr`rerhrlro__static_attributes__ __classcell__rs@r?rrs6NIII E D2</2 <#' 8,3''&   3 3"AA //(/)/(/)/(/)/,(3)3'''',,",0     &&rerc \rSrSr%SrSrS\S'SrSrSr Sr Sr Sr \ R\R!S55rS3S jr\S4S j5rS rS rS rSrSrSr\S5rSrSrSrSr Sr!Sr"Sr#Sr$Sr%Sr&Sr'Sr(\)"S\*5S5r+\)"S\*5S 5r,\)"S\*5S!5r-\)"S\*5S"5r.\)"S\*5S#5r/\)"S\*5S$5r0S%r1S&r2S'r3S(r4S)r5S*r6S+r7S,r8S-r9S.r:S/r;S5S0jrg)6rQiRaaRepresent a floating-point number of arbitrary precision. Examples ======== >>> from sympy import Float >>> Float(3.5) 3.50000000000000 >>> Float(3) 3.00000000000000 Creating Floats from strings (and Python ``int`` and ``long`` types) will give a minimum precision of 15 digits, but the precision will automatically increase to capture all digits entered. >>> Float(1) 1.00000000000000 >>> Float(10**20) 100000000000000000000. >>> Float('1e20') 100000000000000000000. However, *floating-point* numbers (Python ``float`` types) retain only 15 digits of precision: >>> Float(1e20) 1.00000000000000e+20 >>> Float(1.23456789123456789) 1.23456789123457 It may be preferable to enter high-precision decimal numbers as strings: >>> Float('1.23456789123456789') 1.23456789123456789 The desired number of digits can also be specified: >>> Float('1e-3', 3) 0.00100 >>> Float(100, 4) 100.0 Float can automatically count significant figures if a null string is sent for the precision; spaces or underscores are also allowed. (Auto- counting is only allowed for strings, ints and longs). >>> Float('123 456 789.123_456', '') 123456789.123456 >>> Float('12e-3', '') 0.012 >>> Float(3, '') 3. If a number is written in scientific notation, only the digits before the exponent are considered significant if a decimal appears, otherwise the "e" signifies only how to move the decimal: >>> Float('60.e2', '') # 2 digits significant 6.0e+3 >>> Float('60e2', '') # 4 digits significant 6000. >>> Float('600e-2', '') # 3 digits significant 6.00 Notes ===== Floats are inexact by their nature unless their value is a binary-exact value. >>> approx, exact = Float(.1, 1), Float(.125, 1) For calculation purposes, evalf needs to be able to change the precision but this will not increase the accuracy of the inexact value. The following is the most accurate 5-digit approximation of a value of 0.1 that had only 1 digit of precision: >>> approx.evalf(5) 0.099609 By contrast, 0.125 is exact in binary (as it is in base 10) and so it can be passed to Float or evalf to obtain an arbitrary precision with matching accuracy: >>> Float(exact, 5) 0.12500 >>> exact.evalf(20) 0.12500000000000000000 Trying to make a high-precision Float from a float is not disallowed, but one must keep in mind that the *underlying float* (not the apparent decimal value) is being obtained with high precision. For example, 0.3 does not have a finite binary representation. The closest rational is the fraction 5404319552844595/2**54. So if you try to obtain a Float of 0.3 to 20 digits of precision you will not see the same thing as 0.3 followed by 19 zeros: >>> Float(0.3, 20) 0.29999999999999998890 If you want a 20-digit value of the decimal 0.3 (not the floating point approximation of 0.3) you should send the 0.3 as a string. The underlying representation is still binary but a higher precision than Python's float is used: >>> Float('0.3', 20) 0.30000000000000000000 Although you can increase the precision of an existing Float using Float it will not increase the accuracy -- the underlying value is not changed: >>> def show(f): # binary rep of Float ... from sympy import Mul, Pow ... s, m, e, b = f._mpf_ ... v = Mul(int(m), Pow(2, int(e), evaluate=False), evaluate=False) ... print('%s at prec=%s' % (v, f._prec)) ... >>> t = Float('0.3', 3) >>> show(t) 4915/2**14 at prec=13 >>> show(Float(t, 20)) # higher prec, not higher accuracy 4915/2**14 at prec=70 >>> show(Float(t, 2)) # lower prec 307/2**10 at prec=10 The same thing happens when evalf is used on a Float: >>> show(t.evalf(20)) 4915/2**14 at prec=70 >>> show(t.evalf(2)) 307/2**10 at prec=10 Finally, Floats can be instantiated with an mpf tuple (n, c, p) to produce the number (-1)**n*c*2**p: >>> n, c, p = 1, 5, 0 >>> (-1)**n*c*2**p -5 >>> Float((1, 5, 0)) -5.00000000000000 An actual mpf tuple also contains the number of bits in c as the last element of the tuple: >>> _._mpf_ (1, 5, 0, 3) This is not needed for instantiation and is not the same thing as the precision. The mpf tuple and the precision are two separate quantities that Float tracks. In SymPy, a Float is a number that can be computed with arbitrary precision. Although floating point 'inf' and 'nan' are not such numbers, Float can create these numbers: >>> Float('-inf') -oo >>> _.is_Float False Zero in Float only has a single value. Values are not separate for positive and negative zeroes. rwrEztuple[int, int, int, int]rwNTz-+_.c  UbUb [S5e[U[5(aUR5=pAUR SS5R 5nUR S5(a[U5S:aSU-nGO!UR S5(a[U5S:a S USS-nGOUS ;a[R$US :Xa[R$US :Xa[R$[U5(d[S U-5eGO[U[5(a US:XaSnGOq[U[5(aU[S5:Xa[R$[U[5(aU[S 5:Xa[R$[U[5(a+[R"U5(a[R$[U[ ["45(a [U5nOU[RLaU$U[RLaU$U[RLaU$[%U5(a ['U5nOG[U[(R*5(a(UcUcUR,R.nUR0nUcUcSn[U[25(aU$[U[5(an[4R6"U5nSU;n[9U5upUR:(aU(a[=U[?U55n[=SU5n[AU5nOOUS:XaUb UcUS:Xa[U[5(d [S5e[4R6"U5nSU;n[9U5upUR:(a'U(a [=U[?U55n[AU5nUbUS:Xa [AU5n[EU5n[U[5(a[FRH"X[J5nGOl[U[5(a[FRL"X[J5nGO:[U[4R65(aURO5(a'[FRL"[U5U[J5nGOURQ5(a[R$URS5(a&US:a[R$[R$[S[U5-5e[U[T5(Ga[U5S;a[US[5(aO[WU5nUSRYS5R[S5US'[]USS5US'[UU5nO[U5S:Xa[2R_X5$[aUSS;USS:[aSU5545(d[SU<35e[2R_USUSUS[cUS54U5$[U[d[f45(aURiU5nO[(R*"XS9R0nUR_XsSS9$![4RBa GNf=f![4RBa [SU-5ef=f) Nz=Both decimal and binary precision supplied. Supply only one.  r`rr0z-.r5z-0.)rrrrzstring-float not recognized: %srrr8z^The null string can only be used when the number to Float is passed as a string or an integer.z*string-float not recognized by Decimal: %szunexpected decimal value %s)0xLr)rrc3R# UHn[U5[[4;v M g7fr:)rrWr<s r?r@ Float.__new__..sCs!QC: 5ss%'zmalformed mpf: )rkF)zero)5rNrLrMstripreplacerrrr rrrrrmathisnanrrr r ryrjcontextrkrwrQrrrrrrr1InvalidOperationrWr from_floatrifrom_strris_nanrrlist removeprefix removesuffixr%rrOr"r NumberSymbolr)rnumdps precision_numNumisintrws r?r Float.__new__ s ?y412 2 c3   $D++c3'--/C~~c""s3x!|Ci%%#c(Q,c!"go'zz!)))uu #C(( !BT!IJJ) U # #qC U # #uU|(;::  U # #uV}(<%% % U # # 355L j'2 3 3c(C AJJ J A&& &J AEE\J  $ $&s+C VZZ ( ( ; # 0 0I))C ;9,C#u%% #s## 1!//#.C sNE8=HC~~%"#z#7b#,C +C 0I$"_ 0AcRic3'' "KLL 1ooc*34S9>>ec:c?3C +C 0I   R#C(I N c5 ! !OOCC8E S ! !MM##6E W__ - -}} c#h 3?uu ""7::%))) !>S!IJJ U # #CF(:#a&#&&3iQ,,T2??DASVRAc s8q= ::c55Ff,FaKCsCC  ))EFF!::QQQ#a&1AB!##fl3 4 4OOI.EJJs399Exxux55s//"++ U !MPS!STT UsZ(;[([?[#[&c<U(aU[:Xa[R$U[:Xa[R$U[ :Xa[R $U[:Xa[R$[R"U5n[X5Ul X$l U$r:)r.r r'_mpf_nanr_mpf_infr _mpf_ninfrrrrqrwrE)rrwrErrs r?r Float._newsq EUN66M h 55L h ::  i %% %ll3U*   rechURupp4U[U5SSX44nSUR0nU4U4$)Nr5r)rwhexrE)rrlrmexproargrVs r?__getnewargs_ex__Float.__getnewargs_ex__sA!ZZ3SXab\3+tzz*rec2URUR4$r:rrs r?_hashable_contentFloat._hashable_contents DJJ''rec [[[R"[R"UR UR 5555$r:)rrWrto_int mpf_floorrwrErs r?r Float.floors7s4;; NN4::tzz 2456 6rec [[[R"[R"UR UR 5555$r:)rrWrrmpf_ceilrwrErs r?r Float.ceilings7s4;; MM$**djj 1345 5rec"UR5$r:rrs r?rFloat.__floor__rrec"UR5$r:rrs r?rFloat.__ceil__rrecB[R"UR5$r:)ryrjrwrs r?r Float.numszz$**%%rec[URU5nX R:wa&URU:Xa[URU5 U$r:)rqrwrEr2rrkrps r?rFloat._as_mpf_vals: djj$ '   d 2 $**b ! recDUR[XR54$r:)rwrrErs r?rFloat._as_mpf_opszz3tZZ000rec:UR[[4;agg)NFTrwrrrs r?_eval_is_finiteFloat._eval_is_finites ::(I. .rec:UR[[4;aggNTFrrs r?_eval_is_infiniteFloat._eval_is_infinites ::(I. .recPUR[:Xag[U5(dggr)rwr. int_valuedrs r?_eval_is_integerFloat._eval_is_integers$ :: $ recVUR[[4;agURS:$NFrrwrrrrs r?_eval_is_negativeFloat._eval_is_negative$ ::)X. .xx!|recVUR[[4;agURS:$rrrs r?_eval_is_positiveFloat._eval_is_positiverrectUR[:XagUR[:XagURS:$NTFrrrs r?_eval_is_extended_negative Float._eval_is_extended_negatives. :: " :: !xx!|rectUR[:XagUR[:XagURS:$r)rwrrrrs r?_eval_is_extended_positive Float._eval_is_extended_positives. :: ! :: "xx!|rec(UR[:H$r:rwr.rs r? _eval_is_zeroFloat._eval_is_zerozzU""rec(UR[:g$r:rrs r?__bool__Float.__bool__rrecU(dU$[R[R"UR5UR 5$r:)rQrrmpf_negrwrErs r?r Float.__neg__s,Kzz$,,tzz2DJJ??rerc.[U[5(al[R(aWUR UR 5up#[ R[R"URX#[5U5$[RX5$r:) rLrr4rrrErQrrmpf_addrwrirrrrhsrks r?r Float.__add___ eV $ $):)C)C((4IC::dll4::s#FM M~~d**rec.[U[5(al[R(aWUR UR 5up#[ R[R"URX#[5U5$[RX5$r:) rLrr4rrrErQrrmpf_subrwrirrs r?r Float.__sub__ rrec.[U[5(al[R(aWUR UR 5up#[ R[R"URX#[5U5$[RX5$r:) rLrr4rrrErQrrmpf_mulrwrir$rs r?r$ Float.__mul__rrec:[U[5(arUS:wal[R(aWUR UR 5up#[ R[R"URX#[5U5$[RX5$Nr) rLrr4rrrErQrrmpf_divrwrir(rs r?r(Float.__truediv__sg eV $ $!8I8R8R((4IC::dll4::s#FM M!!$..rec[U[5(aVURS:waF[R(a1[ [R [U5U5URS9$[U[ 5(aQ[R(a<X- n[U5(a([ S[URUR5S9$[U[5(al[R(aWURUR5up4[ R[R"URX4[ 5U5$[R X5$)Nrrr)rLrr|r4rrQ__mod__rErrrrrrmpf_modrwri)rrrrrks r?r Float.__mod__s eX & &577a< exp(r*log(-p)) -> exp(r*(log(p) + I*Pi)) -> -> p**r*(sin(Pi*r) + cos(Pi*r)*I) rrr5FrN) equal_valuedis_extended_positiverr ComplexInfinityrLrrrErQrr mpf_pow_intrwrzrirr|rSPowrT _eval_powerrr& ComplexResultmpc_powr. ImaginaryUnit)rrnrkmpfselfyreims r?rFloat._eval_power6s a (( ((((( dF # #$((zzzz$$TZZsCTKKD(++FFaKDFFQJ43C3C1==$?E;;t,--4JDjjG 9G45zz!** $%% 9e$tUmT3@zz"+JJr(899 9s"%F A;HHc~[R[R"UR5UR 5$r:)rQrrmpf_absrwrErs r?__abs__ Float.__abs__Vs$zz$,,tzz2DJJ??rec~UR[:Xag[[R"UR55$r)rwr.rWrrrs r?__int__ Float.__int__Ys) :: 4;;tzz*++recn[U[5(a [U5n[R"X5$r:)rLrrQrr+rs r?r+ Float.__eq__^s' eU # #%LE||D((recHURU5nU[LaU$U(+$r:)r+r)rreqs r?r/ Float.__ne__cs$ [[   I6Mrec[U5n[R"U5(d [U5$[R "U5$r:)rrisinfhashrrD)r float_vals r?rDFloat.__hash__js3$K zz)$$ ? "~~d##rec T[U5nUR(az[R "UR [R"UR55n[R"UR5n[[U"X4555$UR(a/[[U"UR UR 555$UR(aU[R[R4;aUR![#UR$55nUR$S:aPUR&(a>[[U"UR UR)UR$5555$gggg![a [s$f=fNr)r rrrRrrrwfrom_intr|rzris_Float is_comparabler rrevalfr0rErxr)rropsmpfompfs r?_Frel Float._Frelps@ "UOE    << DMM%'',BCD==)DDD01 1 ^^D4::u{{356 6  U A..30&0KK DJJ 78E{{Q??#D4::u'8'8'DE%GHH#&0 % "! ! "s FF'&F'c[U[5(aURU5$URU[R 5nUc[ R"X5$U$r:)rLrr4r*rmpf_gtrr;rrrps r?r; Float.__gt__K e\ * *<<% % ZZt{{ + :;;t+ + rec[U[5(aURU5$URU[R 5nUc[ R"X5$U$r:)rLrr8r*rmpf_gerr?r.s r?r? Float.__ge__r0rec[U[5(aURU5$URU[R 5nUc[ R"X5$U$r:)rLrr;r*rmpf_ltrr4r.s r?r4 Float.__lt__r0rec[U[5(aURU5$URU[R 5nUc[ R"X5$U$r:)rLrr?r*rmpf_lerr8r.s r?r8 Float.__le__r0rec4[X- 5[U5:$r:)rXrQ)rrepsilons r? epsilon_eqFloat.epsilon_eqs4< 5>11recT[[R"[U55U5$r:)formatrrrM)r format_specs r? __format__Float.__format__sgooc$i0+>>rerNNT)z1e-15)?rrsrtrurvry__annotations__ is_rational is_irrationalr;is_realis_extended_realr$rM maketransdictfromkeys_remove_non_digitsrr{rrrrrrrpropertyrrrrrrrrrrrrrrrrrr$r(rrrrrr+r/rDr*r;r?r4r8r<rAr|rrer?rQrQRsdJ#I $$KMIGHt}}V'<=L6\  (65&& 1     ##@ (+)+ (+)+ (+)+ (/)/ ( +) +(,),9@@, ) $ H:2?rerQc^\rSrSr%SrSrSrSrSrSr S\ S'S\ S'Sr \ S6S j5r \S7S j5r\S8S j5rS9S jrS rSrSrSrSr\"S\5S5r\r\"S\5S5r\"S\5S5r\"S\5S5r\r\"S\5S5r\"S\5S5r \"S\5S5r!\"S\5S5r"Sr#Sr$Sr%Sr&Sr'S r(S!r)S"r*S#r+S$r,S%r-S&r.S'r/S(r0S)r1S*r2U4S+jr3S:S,jr4\5S-5r6\5S.5r7\"S\5S/5r8\"S\5S05r9S1r:S;S2jr;SU=r?$)=ria}Represents rational numbers (p/q) of any size. Examples ======== >>> from sympy import Rational, nsimplify, S, pi >>> Rational(1, 2) 1/2 Rational is unprejudiced in accepting input. If a float is passed, the underlying value of the binary representation will be returned: >>> Rational(.5) 1/2 >>> Rational(.2) 3602879701896397/18014398509481984 If the simpler representation of the float is desired then consider limiting the denominator to the desired value or convert the float to a string (which is roughly equivalent to limiting the denominator to 10**12): >>> Rational(str(.2)) 1/5 >>> Rational(.2).limit_denominator(10**12) 1/5 An arbitrarily precise Rational is obtained when a string literal is passed: >>> Rational("1.23") 123/100 >>> Rational('1e-2') 1/100 >>> Rational(".1") 1/10 >>> Rational('1e-2/3.2') 1/320 The conversion of other types of strings can be handled by the sympify() function, and conversion of floats to expressions or simple fractions can be handled with nsimplify: >>> S('.[3]') # repeating digits in brackets 1/3 >>> S('3**2/10') # general expressions 9/10 >>> nsimplify(.3) # numbers that have a simple form 3/10 But if the input does not reduce to a literal Rational, an error will be raised: >>> Rational(pi) Traceback (most recent call last): ... TypeError: invalid input: pi Low-level --------- Access numerator and denominator as .p and .q: >>> r = Rational(3, 4) >>> r 3/4 >>> r.p 3 >>> r.q 4 Note that p and q return integers (not SymPy Integers) so some care is needed when using them in expressions: >>> r.p/r.q 0.75 See Also ======== sympy.core.sympify.sympify, sympy.simplify.simplify.nsimplify TFrzr|rWrzr|cUGci[U[5(aU$[U[5(aGO9[U[[45(a[[ U56$[U[ 5(d [U5nOURS5S:a[SU-5eURSS5nURSS5n[U5S:Xa4Uup[R "U5n[R "U5nXV- n[R "U5nUR#UR$UR&S5$[U[5(d[SU-5eSnSn[U[5(d&[U5nXqR*-nUR,nO [/U5n[U[5(d([U5nXR*-nXrR,-nOU[/U5-nUnUb[1SSS S S 9 UR#XU5$UR#X5$![[4a Nf=f![(a GNf=f) N/rzinvalid input: %srr8r5z4gcd is deprecated in Rational, use nsimplify insteadz1.11zdeprecated-rational-gcdr)deprecated_since_versionactive_deprecations_target stacklevel)rLrrrrQr}rMr r SyntaxErrorcountrrrsplitr fractionsFractionr numerator denominatorrNr|rzrWr3)rrzr|rhpqfpfqQs r?rRational.__new__s 9!X&&!Z((a%00#%6q%9::!!S))#AJwws|a''(;a(?@@ #r*A#q)B2w!|!&//2&//2EG%..q1 #xx Q]]AFF!!X..#$7!$;<<A !Z(( A HAAAA!Z(( A HA HA QKA  ? %F)/+D   88A#& &xx~g)+6&s$5 H5 I 5II IIcUS:Xa?US:Xa)[S(a [S5e[R$[R$US:aU*nU*nUc[ [ U5U5nUS:aX-nX#-nURX5$)NrrrzIndeterminate 0/0r)rtrNr rrrrXfrom_coprime_ints)rrzr|rhs r?r Rational._new^s 6AvH%$%89955L$$ $ q5AA ;s1vq/C 7 IA IA$$Q**recUS:Xa [U5$US:XaUS:Xa[R$[R"U5nXlX#lU$)auCreate a Rational from a pair of coprime integers. Both ``p`` and ``q`` should be strictly of type ``int``. The caller should ensure that ``gcd(p,q) == 1`` and ``q > 0``. This may be more efficient than ``Rational(p, q)``. The validity of the arguments may or may not be checked so it should not be relied upon to pass unvalidated or invalid arguments to this function. rr5)rr Halfrrrzr|)rrzr|rs r?rcRational.from_coprime_intsusH 61:  6a1f66Mll3 rec [R"URUR5n[ UR [R"[ U5555$)zClosest Rational to self with denominator at most max_denominator. Examples ======== >>> from sympy import Rational >>> Rational('3.141592653589793').limit_denominator(10) 22/7 >>> Rational('3.141592653589793').limit_denominator(100) 311/99 )rYrZrzr|rlimit_denominatorrW)rmax_denominatorrs r?riRational.limit_denominatorsD   tvvtvv .++I,>,>s??S,TUVVrec2URUR4$r:rPrs r?__getnewargs__Rational.__getnewargs__rec2URUR4$r:rPrs r?rRational._hashable_contentrorec URS:$rrzrs r?rRational._eval_is_positivesvvzrec URS:H$rrsrs r?rRational._eval_is_zerosvv{recD[UR*UR5$r:)rrzr|rs r?rRational.__neg__s((rerc8[R(a[U[5(aE[R UR URUR --URS5$[U[5(aT[ UR UR-URUR --URUR-5$[U[5(aX-$[RX5$[RX5$r") r4rrLrrrrzr|rQrrrs r?rRational.__add__s  % %%))}}TVVdffUWWn%[R(a[U[5(aURUR -URUR --n[URUR -X!R-UR -- UR UR -5$[U[ 5(a,[ UR[U55URS9$[RX5$[RX5$)Nr) r4rrLrrzr|rQrrEr)rrrSs r?rRational.__mod__s  % %%**VVEGG^8uww774661A A466%''>RR%''T\\(5/:',{{44>>$. .~~d**rec[U[5(a[RX5$[R X5$r:)rLrrrrrs r?rRational.__rmod__s/ eX & &##E0 0t++rec[U[5(Ga[U[5(aURUR5U-$UR (aU*nU[ RLa [URUR5$UR(a8[ RU-[URUR*5U--$[URUR5U-$U[ RLaURUR:a[ R$URUR*:a2[ R[ R[ R--$[ R$[U[ 5(aE[R#URUR-URUR-S5$[U[5(GaURUR-nU(aUS- nX1R-UR- n[XAR5nURS:waV[!UR5U-[!UR5U--[R#SURU-S5-$[!UR5U-[R#SURU-S5-$URUR- n[XAR5nURS:waS[!UR5U-[!UR5U--[R#SURS5-$[!UR5U-[R#SURS5-$UR (aUR$(aU*U-$gr")rLrrQrrErr rrr|rzrSrTrr r'rris_even)rrnneintpart remfracpart ratfracparts r?rRational._eval_powers dF # #$&&'' 3T99((U!%%K#DFFDFF33##==$.x/H"/LLL#DFFDFF3R77qzz!66DFF?::%66TVVG#:: 1??(BBBvv $((}}TVVTVV^TVVTVV^QGG$))&&DFF*qLG")&&.466"9K"*;"?Kvv{&tvv4WTVV_k5QQRZR_R_`acgcicikrcrtuRvvv"466?K7 aQXZ[8\\\"&&&466/K"*;"?Kvv{&tvv4WTVV_k5QQRZR_R_`acgciciklRmmm"466?K7 aQR8SSS  $ $ED= recd[R"URURU[5$r:)r from_rationalrzr|rirs r?rRational._as_mpf_val;s!!!$&&$&&$<s(t11$&&$&&$LMMrecT[[UR5UR5$r:)rrXrzr|rs r?rRational.__abs__AsDFF TVV,,recvURURp!US:a[U*U-5*$[X-5$r)rzr|rW)rrzr|s r?rRational.__int__Ds6vvtvv1 q5AJ; 14yrecF[URUR-5$r:rrzr|rs r?rRational.floorJstvv'((recJ[UR*UR-5*$r:rrs r?rRational.ceilingMs466)***rec"UR5$r:rrs r?rRational.__floor__Prrec"UR5$r:rrs r?rRational.__ceil__Srrecp[U5n[U[5(dgUR (a#UR (agURU5$UR(a9URUR:H=(a URUR:H$g![a [s$f=fNF) r rrrLris_NumberSymbolrGr+rRrzr|rs r?r+Rational.__eq__Vs "UOE%((  ""<<% %   66UWW$:577): : "! ! "s B""B54B5c X:X+$r:rrs r?r/Rational.__ne__i   recR[U5nUR(aSnXpTUR(a [ XR5nO|UR (a [ XR5nO_UR(aN[URUR-5[URUR-5pT[ XR5nU(aU"U5$UR(a6UR(a$[UR5URU-4$ggg![a [s$f=fr:) r rrrxrrVr$rRrrzr|r;rI)rrattrr'ros r?_RrelRational._Rrells "UOE ??Bq$$Q%Q%""qss133w'QSS)91Q%!u {{q11qss|QSSU** 2{  "! ! "s DD&%D&cURUS5nUcX4nO[U[5(dU$[R"U6$)Nr4)rrLrrr;r.s r?r;Rational.__gt__? ZZx ( :BB&&I{{BrecURUS5nUcX4nO[U[5(dU$[R"U6$)Nr8)rrLrrr?r.s r?r?Rational.__ge__rrecURUS5nUcX4nO[U[5(dU$[R"U6$)Nr;)rrLrrr4r.s r?r4Rational.__lt__rrecURUS5nUcX4nO[U[5(dU$[R"U6$)Nr?)rrLrrr8r.s r?r8Rational.__le__rrec >[TU]5$r:rBrEs r?rDRational.__hash__rGrec <SSKJn U"XUX4US9R5$)zA wrapper to factorint which return factors of self that are smaller than limit (or cheap to compute). Special methods of factoring are disabled by default so that only trial division is used. r) factorrat)limit use_trialuse_rhouse_pm1verbose)sympy.ntheory.factor_rcopy)rrrrrrvisualrs r?factorsRational.factorss% 4i%%''+tv .recUR$r:rsrs r?r[Rational.numerator vv recUR$r:)r|rs r?r\Rational.denominatorrrec[U[5(a^U[R:XaU$[[ UR UR 5[ URUR55$[RX5$r:) rLrr r'rrzrr|rrhrs r?rh Rational.gcdsa eX & & TVVUWW%TVVUWW%' 'zz$&&rec[U[5(ab[UR[URUR5-UR-[URUR55$[ R X5$r:)rLrrzrr|rrlrs r?rl Rational.lcmsc eX & &$tvvuww//%''9TVVUWW%' 'zz$&&recV[UR5[UR54$r:rrs r?as_numer_denomRational.as_numer_denomstvv//recU(a6UR(aU[R4$U*[R4$[RU4$)zReturn the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import S >>> (S(-3)/2).as_content_primitive() (3/2, -1) See docstring of Expr.as_content_primitive for more examples. )r#r rrT)rradicalclears r?as_content_primitiveRational.as_content_primitives= QUU{"5!--' 'uud{rec&U[R4$r]r^r_s r?r`Rational.as_coeff_MulsQUU{rec&U[R4$rcrdr_s r?reRational.as_coeff_AddsQVV|rerrCr:)rzrWr|rWrqr)i@BNTFFFF)FTrr)@rrsrtrurvrH is_integerrFr;ryrErRrrr{rrcrirmrrrrrrr__radd__rr~r$__rmul__r(rrrrrrrrrrrrr+r/rr;r?r4r8rDrrNr[r\rhrlrrr`rer|r}r~s@r?rrsNQdGJKII F FK A AF++,*W   )( +) +H( +) +( ,) ,( +) +H( /) /( 0) 0( +) +(,), +Z=N- )+&!+*    ";@5: .(')'(')'0(rercv^\rSrSrSrSrSrSrSrSr Sr Sr \ S5r S rS rS rS rS rSrSrSrSrSrSrSrSrSrSrSrSrSrSr Sr!Sr"Sr#Sr$S r%S!r&S"r'S#r(U4S$jr)S%r*S&r+S'r,\-"S(\.5S)5r/S*r0S+r1S,r2S-r3S.r4S/r5S0r6S1r7S2r8S3r9S4r:S5r;Sr>> from sympy import Integer >>> Integer(3) 3 If a float or a rational is passed to Integer, the fractional part will be discarded; the effect is of rounding toward zero. >>> Integer(3.8) 3 >>> Integer(-3.8) -3 A string is acceptable input if it can be parsed as an integer: >>> Integer("9" * 20) 99999999999999999999 It is rarely needed to explicitly instantiate an Integer, because Python integers are automatically converted to Integer when they are used in SymPy expressions. rTrcN[R"URU[5$r:)rr#rzrirs r?rInteger._as_mpf_val s}}TVVT3//recL[R"URU55$r:)ryrrrs r?rInteger._mpmath_st//566rec`[U[5(aURSS5n[U5nUS:Xa[ R $US:Xa[ R$US:Xa[ R$[R"U5nX#l U$![a [ SU-5ef=f)Nrr8z6Argument of Integer should be of numeric type, got %s.rrxr) rLrMrrWrr rrTr'rrrz)rr>ivalrs r?rInteger.__new__s a   #r"A Nq6D 1955L 2:== 1966Mll3  NH1LN N Ns BB-cUR4$r:rsrs r?rmInteger.__getnewargs__.syrecUR$r:rsrs r?rInteger.__int__2 vv rec,[UR5$r:rrzrs r?r Integer.floor5tvvrec,[UR5$r:rrs r?rInteger.ceiling8rrec"UR5$r:rrs r?rInteger.__floor__;rrec"UR5$r:rrs r?rInteger.__ceil__>rrec.[UR*5$r:rrs r?rInteger.__neg__AwrecRURS:aU$[UR*5$r)rzrrs r?rInteger.__abs__Ds# 66Q;KDFF7# #rec[U[5(a;[R(a&[ [ UR UR 56$[RX5$r:) rLrr4rrrrzrrrs r?rInteger.__divmod__JsC eW % %*;*D*D6$&&%''24 4$$T1 1rec`[U[5(a0[R(a[ [ XR 56$[U5n[RX5$![a< Sn[U5Rn[U5Rn[X#U4-5ef=f)Nz7unsupported operand type(s) for divmod(): '%s' and '%s') rLrWr4rrrrzrrrrr)rrronamesnames r?rInteger.__rdivmod__Ps eS ! !&7&@&@6%02 2 6u  $$U1 1  6OU ,,T ++en 455  6s  A''AB-c[R(a[U[5(a[ UR U-5$[U[5(a"[ UR UR -5$[U[ 5(aE[ RUR UR-UR -URS5$[ RX5$[X5$r") r4rrLrWrrzrrr|rAddrs r?rInteger.__add__^s  % %%%%tvv~..E7++tvv/00E8,,}}TVVEGG^egg%=uwwJJ##D0 0t# #rec[R(a[U[5(a[ XR -5$[U[ 5(aE[ RUR UR UR--URS5$[ RX5$[ RX5$r") r4rrLrWrrzrrr|rrs r?rInteger.__radd__j  % %%%%uvv~..E8,,}}UWWtvvegg~%=uwwJJ$$T1 1  --rec[R(a[U[5(a[ UR U- 5$[U[5(a"[ UR UR - 5$[U[ 5(aE[ RUR UR-UR - URS5$[ RX5$[ RX5$r") r4rrLrWrrzrrr|rrs r?rInteger.__sub__ss  % %%%%tvv~..E7++tvv/00E8,,}}TVVEGG^egg%=uwwJJ##D0 0,,rec[R(a[U[5(a[ XR - 5$[U[ 5(aE[ RUR UR UR-- URS5$[ RX5$[ RX5$r") r4rrLrWrrzrrr|r~rs r?r~Integer.__rsub__~r rec[R(a[U[5(a[ UR U-5$[U[5(a"[ UR UR -5$[U[ 5(aV[ RUR UR -UR[UR UR55$[ RX5$[ RX5$r:) r4rrLrWrrzrrr|rr$rs r?r$Integer.__mul__s  % %%%%tvve|,,E7++tvvegg~..E8,,}}TVVEGG^UWWd466577>STT##D0 0,,rec[R(a[U[5(a[ XR -5$[U[ 5(aV[ RUR UR -UR[UR UR55$[ RX5$[ RX5$r:) r4rrLrWrrzrrr|rrrs r?rInteger.__rmul__s  % %%%%uVV|,,E8,,}}UWWTVV^UWWd466577>STT$$T1 1  --recH[R(ay[U[5(a[ UR U-5$[U[5(a"[ UR UR -5$[ RX5$[ RX5$r:)r4rrLrWrrzrrrs r?rInteger.__mod__sr  % %%%%tvv~..E7++tvv/00##D0 0,,recF[R(ax[U[5(a[ XR -5$[U[5(a"[ UR UR -5$[ RX5$[ RX5$r:)r4rrLrWrrzrrrs r?rInteger.__rmod__sp  % %%%%uvv~..E7++uww/00$$T1 1  --rec[U[5(aURU:H$[U[5(aURUR:H$[R X5$r:)rLrWrzrrr+rs r?r+Integer.__eq__sL eS ! !FFeO $ w ' 'FFegg% &t++rec X:X+$r:rrs r?r/Integer.__ne__rrec[U5nUR(a"[URUR:5$[ R X5$![a [s$f=fr:)r rrrrzrr;rs r?r;Integer.__gt__Y "UOE   DFFUWW,- -t++  "! ! " AA('A(c[U5nUR(a"[URUR:5$[ R X5$![a [s$f=fr:)r rrrrzrr4rs r?r4Integer.__lt__rrc[U5nUR(a"[URUR:5$[ R X5$![a [s$f=fr:)r rrrrzrr?rs r?r?Integer.__ge__Y "UOE   DFFegg-. .t++  "! ! "rc[U5nUR(a"[URUR:*5$[ R X5$![a [s$f=fr:)r rrrrzrr8rs r?r8Integer.__le__r#rc,[UR5$r:)rrzrs r?rDInteger.__hash__sDFF|recUR$r:rsrs r? __index__Integer.__index__rrec2[URS-5$Nr5)rrzrs r? _eval_is_oddInteger._eval_is_oddsDFFQJrec >SSKJn U[RLa`UR[R :a[R$[R[R [R--$U[RLa([RSUS5[R-$[U[5(d(UR(aUR(aU*U-$[U[5(a[TU]AU5$[U[5(dgU[R"La/UR(a[R [%U*U5-$UR(aqU*nUR(a9[R&U-[RSUR*S5U--$[RSURS5U-$[)[+UR5UR,5upEU(aJ[/U[+UR5-5nUR(aU[R&U--nU$[1[+UR55nU"U5nUSLa[1US5[1US50n O[/U5R3SS9n Sn Sn Sn Sn 0nU R55HunnUUR-n[7UUR,5unnUS:aXU--n US:dMC[9UUR,5nUS:wa6U [%U[RUU-UR,U-S55-n MUX'M UR55H$unnU S:XaUn M[9U U5n U S:XdM$ O UR55HunnU UUU ---n M X:XaU S:Xa U S:XaSnU$X-[%U [XR,55-nUR(aU[%[R&U5-nU$)a Tries to do some simplifications on self**expt Returns None if no further simplifications can be done. Explanation =========== When exponent is a fraction (so we have for example a square root), we try to find a simpler representation by factoring the argument up to factors of 2**15, e.g. - sqrt(4) becomes 2 - sqrt(-4) becomes 2*I - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7) Further simplification would require a special call to factorint on the argument which is not done here for sake of speed. r) perfect_powerrNFi)r)rr0r rrzrr rrrrLrrSrrQrCrrfrrTrrXr|rrWritemsrr)rrnr0rxxexactresultb_posrzrKout_intout_radsqr_intsqr_gcdsqr_dictprimeexponentdiv_ediv_mrexkvrs r?rInteger._eval_powersx* 8 1:: vv~zz!:: :: : 1%% %==D!,ajj8 8$''DLL}$ dE " "7&t, ,$))  166>d..??3ud#33 3   B}}d*8==TVVGQ+G+KKK}}Q2B66#CK8  QDFF ^,F!----M CK  %  E>!Is1Q4y)D5>)))6D#zz|OE8  H!(DFF3LE5qy%<'qy'6s5(--q$&&!)Q*OPPG&+HO ,^^%EAr!|w+a< &NN$DAq q1g: &G%  1 AF  _S(7FF2K%LLF#ammT22 recSSKJn U"U5$)Nr)isprime)sympy.ntheory.primetestrD)rrDs r?_eval_is_primeInteger._eval_is_primeTs3t}rec:US:a[UR5$g)NrF)ris_primers r?_eval_is_compositeInteger._eval_is_compositeYs !8T]]+ +rec&U[R4$r:r^rs r?rInteger.as_numer_denom_sQUU{rerc[U[5(d[$[U[5(a[URU-5$[ X5S$r)rLrrrrzrrs r? __floordiv__Integer.__floordiv__bsE%&&! ! eW % %466U?+ +d"1%%recX[[U5RUR-5$r:rrs r? __rfloordiv__Integer.__rfloordiv__js wu~''466122rec[U[[[R45(a![UR [U5-5$[ $r:rLrWrnumbersIntegralrzrrs r? __lshift__Integer.__lshift__t: ec7G,<,<= > >466SZ/0 0! !rec[U[[R45(a![ [U5UR -5$[ $r:rLrWrVrWrrzrrs r? __rlshift__Integer.__rlshift__z8 ec7#3#34 5 53u:/0 0! !rec[U[[[R45(a![UR [U5- 5$[ $r:rUrs r? __rshift__Integer.__rshift__rZrec[U[[R45(a![ [U5UR - 5$[ $r:r\rs r? __rrshift__Integer.__rrshift__r_rec[U[[[R45(a![UR [U5-5$[ $r:rUrs r?__and__Integer.__and__: ec7G,<,<= > >466CJ./ /! !rec[U[[R45(a![ [U5UR -5$[ $r:r\rs r?__rand__Integer.__rand__8 ec7#3#34 5 53u:./ /! !rec[U[[[R45(a![UR [U5- 5$[ $r:rUrs r?__xor__Integer.__xor__rirec[U[[R45(a![ [U5UR - 5$[ $r:r\rs r?__rxor__Integer.__rxor__rmrec[U[[[R45(a![UR [U5-5$[ $r:rUrs r?__or__Integer.__or__rirec[U[[R45(a![ [U5UR -5$[ $r:r\rs r?__ror__Integer.__ror__rmrec.[UR)5$r:rrs r? __invert__Integer.__invert__rre)>rrsrtrurvr|rr;rryrrrrrmrrrrrrrrrrrrr~r$rrrr+r/r;r4r?r8rDr)r-rrFrJrrrrOrRrXr]rardrgrkrorrrurxr{r|r}r~s@r?rrs:4 AJIJI07  4 $ 2 2 $. -. -.-.,!,,,,  l\  (&)&3" " " " " " " " " "   rerc^\rSrSr%SrSrSrSrSr\ r \ "5r S\ S'SSjrU4SjrS r\S 5rSS jrSS jrS rSrSrSrSr\S5rSrSSjrSrSSjrSrU=r $)AlgebraicNumberia Class for representing algebraic numbers in SymPy. Symbolically, an instance of this class represents an element $\alpha \in \mathbb{Q}(\theta) \hookrightarrow \mathbb{C}$. That is, the algebraic number $\alpha$ is represented as an element of a particular number field $\mathbb{Q}(\theta)$, with a particular embedding of this field into the complex numbers. Formally, the primitive element $\theta$ is given by two data points: (1) its minimal polynomial (which defines $\mathbb{Q}(\theta)$), and (2) a particular complex number that is a root of this polynomial (which defines the embedding $\mathbb{Q}(\theta) \hookrightarrow \mathbb{C}$). Finally, the algebraic number $\alpha$ which we represent is then given by the coefficients of a polynomial in $\theta$. )reprootaliasminpoly _own_minpolyTz set[Basic] free_symbolsc SSKJnJn SSKJn [ U5nSnSn [ U[[45(a$UupU R(dSSK J n U "U 5n OZUR(a2URURURUR 4uppOU"XR#S5SS9UpU R%5n Ubm[ X%5(d%UR'[ U5SU 5n[U6nOYUR)UR+5SU 5n[UR+56nO!UR)S S/SU 5n[S S5nUbGSS KJn U"UR+5UR+5U 5nUR)USU 5n[U6nUR15U R15:aUR3U R5nX4nU=(d U nUb%S S KJn [ UU5(dU"U5nUU4-n[8R:"U/UQ76nUUlU UlUUlU Ul SUlU$) aE Construct a new algebraic number $\alpha$ belonging to a number field $k = \mathbb{Q}(\theta)$. There are four instance attributes to be determined: =========== ============================================================================ Attribute Type/Meaning =========== ============================================================================ ``root`` :py:class:`~.Expr` for $\theta$ as a complex number ``minpoly`` :py:class:`~.Poly`, the minimal polynomial of $\theta$ ``rep`` :py:class:`~sympy.polys.polyclasses.DMP` giving $\alpha$ as poly in $\theta$ ``alias`` :py:class:`~.Symbol` for $\theta$, or ``None`` =========== ============================================================================ See Parameters section for how they are determined. Parameters ========== expr : :py:class:`~.Expr`, or pair $(m, r)$ There are three distinct modes of construction, depending on what is passed as *expr*. **(1)** *expr* is an :py:class:`~.AlgebraicNumber`: In this case we begin by copying all four instance attributes from *expr*. If *coeffs* were also given, we compose the two coeff polynomials (see below). If an *alias* was given, it overrides. **(2)** *expr* is any other type of :py:class:`~.Expr`: Then ``root`` will equal *expr*. Therefore it must express an algebraic quantity, and we will compute its ``minpoly``. **(3)** *expr* is an ordered pair $(m, r)$ giving the ``minpoly`` $m$, and a ``root`` $r$ thereof, which together define $\theta$. In this case $m$ may be either a univariate :py:class:`~.Poly` or any :py:class:`~.Expr` which represents the same, while $r$ must be some :py:class:`~.Expr` representing a complex number that is a root of $m$, including both explicit expressions in radicals, and instances of :py:class:`~.ComplexRootOf` or :py:class:`~.AlgebraicNumber`. coeffs : list, :py:class:`~.ANP`, None, optional (default=None) This defines ``rep``, giving the algebraic number $\alpha$ as a polynomial in $\theta$. If a list, the elements should be integers or rational numbers. If an :py:class:`~.ANP`, we take its coefficients (using its :py:meth:`~.ANP.to_list()` method). If ``None``, then the list of coefficients defaults to ``[1, 0]``, meaning that $\alpha = \theta$ is the primitive element of the field. If *expr* was an :py:class:`~.AlgebraicNumber`, let $g(x)$ be its ``rep`` polynomial, and let $f(x)$ be the polynomial defined by *coeffs*. Then ``self.rep`` will represent the composition $(f \circ g)(x)$. alias : str, :py:class:`~.Symbol`, None, optional (default=None) This is a way to provide a name for the primitive element. We described several ways in which the *expr* argument can define the value of the primitive element, but none of these methods gave it a name. Here, for example, *alias* could be set as ``Symbol('theta')``, in order to make this symbol appear when $\alpha$ is printed, or rendered as a polynomial, using the :py:meth:`~.as_poly()` method. Examples ======== Recall that we are constructing an algebraic number as a field element $\alpha \in \mathbb{Q}(\theta)$. >>> from sympy import AlgebraicNumber, sqrt, CRootOf, S >>> from sympy.abc import x Example (1): $\alpha = \theta = \sqrt{2}$ >>> a1 = AlgebraicNumber(sqrt(2)) >>> a1.minpoly_of_element().as_expr(x) x**2 - 2 >>> a1.evalf(10) 1.414213562 Example (2): $\alpha = 3 \sqrt{2} - 5$, $\theta = \sqrt{2}$. We can either build on the last example: >>> a2 = AlgebraicNumber(a1, [3, -5]) >>> a2.as_expr() -5 + 3*sqrt(2) or start from scratch: >>> a2 = AlgebraicNumber(sqrt(2), [3, -5]) >>> a2.as_expr() -5 + 3*sqrt(2) Example (3): $\alpha = 6 \sqrt{2} - 11$, $\theta = \sqrt{2}$. Again we can build on the previous example, and we see that the coeff polys are composed: >>> a3 = AlgebraicNumber(a2, [2, -1]) >>> a3.as_expr() -11 + 6*sqrt(2) reflecting the fact that $(2x - 1) \circ (3x - 5) = 6x - 11$. Example (4): $\alpha = \sqrt{2}$, $\theta = \sqrt{2} + \sqrt{3}$. The easiest way is to use the :py:func:`~.to_number_field()` function: >>> from sympy import to_number_field >>> a4 = to_number_field(sqrt(2), sqrt(2) + sqrt(3)) >>> a4.minpoly_of_element().as_expr(x) x**2 - 2 >>> a4.to_root() sqrt(2) >>> a4.primitive_element() sqrt(2) + sqrt(3) >>> a4.coeffs() [1/2, 0, -9/2, 0] but if you already knew the right coefficients, you could construct it directly: >>> a4 = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1)/2, 0, S(-9)/2, 0]) >>> a4.to_root() sqrt(2) >>> a4.primitive_element() sqrt(2) + sqrt(3) Example (5): Construct the Golden Ratio as an element of the 5th cyclotomic field, supposing we already know its coefficients. This time we introduce the alias $\zeta$ for the primitive element of the field: >>> from sympy import cyclotomic_poly >>> from sympy.abc import zeta >>> a5 = AlgebraicNumber(CRootOf(cyclotomic_poly(5), -1), ... [-1, -1, 0, 0], alias=zeta) >>> a5.as_poly().as_expr() -zeta**3 - zeta**2 >>> a5.evalf() 1.61803398874989 (The index ``-1`` to ``CRootOf`` selects the complex root with the largest real and imaginary parts, which in this case is $\mathrm{e}^{2i\pi/5}$. See :py:class:`~.ComplexRootOf`.) Example (6): Building on the last example, construct the number $2 \phi \in \mathbb{Q}(\phi)$, where $\phi$ is the Golden Ratio: >>> from sympy.abc import phi >>> a6 = AlgebraicNumber(a5.to_root(), coeffs=[2, 0], alias=phi) >>> a6.as_poly().as_expr() 2*phi >>> a6.primitive_element().evalf() 1.61803398874989 Note that we needed to use ``a5.to_root()``, since passing ``a5`` as the first argument would have constructed the number $2 \phi$ as an element of the field $\mathbb{Q}(\zeta)$: >>> a6_wrong = AlgebraicNumber(a5, coeffs=[2, 0]) >>> a6_wrong.as_poly().as_expr() -2*zeta**3 - 2*zeta**2 >>> a6_wrong.primitive_element().evalf() 0.309016994374947 + 0.951056516295154*I r)ANPDMP)minimal_polynomialNPolygenTpolysr) dup_compose)Symbol)sympy.polys.polyclassesrrsympy.polys.numberfieldsrr rLrris_Polyrris_AlgebraicNumberrrrrget get_domainfrom_sympy_list from_listto_listsympy.polys.densetoolsrdegreeremsymbolrrrr)rexprcoeffsrrrrrrep0alias0rrrdomrscoeffsrcsargsrrs r?rAlgebraicNumber.__new__sR 5?t} dUEN + + MG??6w-  $ $+/<<+/88TZZ+A 'G4/hhuoT348  "  f**))'&/1cB.mmFNN$4a=!12--A3/CAqkG   :CKKM4<<>3?A--1c*CQiG ::<7>>+ +'''++&C   &eV,,u UH$Ell3''   rec >[TU]5$r:rBrEs r?rDAlgebraicNumber.__hash__ rGrec@UR5RU5$r:)as_expr_evalfrs r?rAlgebraicNumber._eval_evalf s||~$$T**recURSL$)z'Returns ``True`` if ``alias`` was set. N)rrs r? is_aliasedAlgebraicNumber.is_aliased szz%%recSSKJnJn UbURURU5$UR b&URURUR 5$SSKJn URURU"S55$)z&Create a Poly instance from ``self``. r)rPurePolyrDummyr2)rrrr rrrr)rr2rrrs r?as_polyAlgebraicNumber.as_poly s^8 =88DHHa( (zz%xx$**55)||DHHeCj99recURU=(d UR5R5R5$)z)Create a Basic expression from ``self``. )rrrexpand)rr2s r?rAlgebraicNumber.as_expr s+||AN+335<<>>recURR5Vs/sH'oRRRU5PM) sn$s snf)z7Returns all SymPy coefficients of an algebraic number. )r all_coeffsrto_sympyrrs r?rAlgebraicNumber.coeffs s:37883F3F3HJ3Ha&&q)3HJJJs.Ac6URR5$)z8Returns all native coefficients of an algebraic number. )rrrs r? native_coeffsAlgebraicNumber.native_coeffs sxx""$$recvSSKJn URnUR5S:XaU$UR5UR 5S- -nUR U"UR UR5- 55nXC-nUR5UR-n[XV4UR55$)z*Convert ``self`` to an algebraic integer. rrr) rrrLCrcomposerrr~r)rrrcoeffpolyrrs r?to_algebraic_integer$AlgebraicNumber.to_algebraic_integer s. LL 446Q;Ka(yyaeeADDFl+,*ttvdii >>rec bSSKJn SSKJn USUSpTURR 5Vs/sHofR U:wdMUPM snHTnU"UR U- 5R(dM)U"U5XT"UR 5-:dMI[U5s $ U$s snf)Nr)CRootOfrmeasureratio) sympy.polys.rootoftoolsr sympy.polysr all_rootsfuncr is_Symbolr~)rrVrrrrrs r?_eval_simplifyAlgebraicNumber._eval_simplify s3' *F7O!\\335K579J!5KAtyy1}%///1:gdii&8 88*1-- L  Ls B, B,cV[URUR4XRS9$)at Form another element of the same number field. Explanation =========== If we represent $\alpha \in \mathbb{Q}(\theta)$, form another element $\beta \in \mathbb{Q}(\theta)$ of the same number field. Parameters ========== coeffs : list, :py:class:`~.ANP` Like the *coeffs* arg to the class :py:meth:`constructor<.AlgebraicNumber.__new__>`, defines the new element as a polynomial in the primitive element. If a list, the elements should be integers or rational numbers. If an :py:class:`~.ANP`, we take its coefficients (using its :py:meth:`~.ANP.to_list()` method). Examples ======== >>> from sympy import AlgebraicNumber, sqrt >>> a = AlgebraicNumber(sqrt(5), [-1, 1]) >>> b = a.field_element([3, 2]) >>> print(a) 1 - sqrt(5) >>> print(b) 2 + 3*sqrt(5) >>> print(b.primitive_element() == a.primitive_element()) True See Also ======== AlgebraicNumber )rr)r~rrr)rrs r? field_elementAlgebraicNumber.field_element s*P \\499 %fJJH Hrec\UR5nUSS/:H=(d XR/:H$)z} Say whether this algebraic number $\alpha \in \mathbb{Q}(\theta)$ is equal to the primitive element $\theta$ for its field. rr)rrrs r?is_primitive_element$AlgebraicNumber.is_primitive_element) s+ KKMQF{.aII;..recNUR(aU$URSS/5$)z Get the primitive element $\theta$ for the number field $\mathbb{Q}(\theta)$ to which this algebraic number $\alpha$ belongs. Returns ======= AlgebraicNumber rr)rrrs r?primitive_element!AlgebraicNumber.primitive_element3 s'  $ $K!!1a&))rec~UR(aU$UR5nURUS9n[X#45$)a Convert ``self`` to an :py:class:`~.AlgebraicNumber` instance that is equal to its own primitive element. Explanation =========== If we represent $\alpha \in \mathbb{Q}(\theta)$, $\alpha \neq \theta$, construct a new :py:class:`~.AlgebraicNumber` that represents $\alpha \in \mathbb{Q}(\alpha)$. Examples ======== >>> from sympy import sqrt, to_number_field >>> from sympy.abc import x >>> a = to_number_field(sqrt(2), sqrt(2) + sqrt(3)) The :py:class:`~.AlgebraicNumber` ``a`` represents the number $\sqrt{2}$ in the field $\mathbb{Q}(\sqrt{2} + \sqrt{3})$. Rendering ``a`` as a polynomial, >>> a.as_poly().as_expr(x) x**3/2 - 9*x/2 reflects the fact that $\sqrt{2} = \theta^3/2 - 9 \theta/2$, where $\theta = \sqrt{2} + \sqrt{3}$. ``a`` is not equal to its own primitive element. Its minpoly >>> a.minpoly.as_poly().as_expr(x) x**4 - 10*x**2 + 1 is that of $\theta$. Converting to a primitive element, >>> a_prim = a.to_primitive_element() >>> a_prim.minpoly.as_poly().as_expr(x) x**2 - 2 we obtain an :py:class:`~.AlgebraicNumber` whose ``minpoly`` is that of the number itself. Parameters ========== radicals : boolean, optional (default=True) If ``True``, then we will try to return an :py:class:`~.AlgebraicNumber` whose ``root`` is an expression in radicals. If that is not possible (or if *radicals* is ``False``), ``root`` will be a :py:class:`~.ComplexRootOf`. Returns ======= AlgebraicNumber See Also ======== is_primitive_element radicals)rminpoly_of_elementto_rootr~)rrrrs r?to_primitive_element$AlgebraicNumber.to_primitive_elementB s>B  $ $K  # # % LL(L +v&&recURc_UR(aURUlUR$SSKJn UR 5nU"UR U5SS9UlUR$)a; Compute the minimal polynomial for this algebraic number. Explanation =========== Recall that we represent an element $\alpha \in \mathbb{Q}(\theta)$. Our instance attribute ``self.minpoly`` is the minimal polynomial for our primitive element $\theta$. This method computes the minimal polynomial for $\alpha$. rrTr)rrr sympy.polys.numberfields.minpolyrr)rrthetas r?r"AlgebraicNumber.minpoly_of_element sk    $(($(LL!    E..0$+DLL,?t$L!   recZUR(a+[UR[5(d UR$U=(d UR 5nUR US9n[ U5S:XaUS$UR5nUHnURXe5(dMUs $ g)a3 Convert to an :py:class:`~.Expr` that is not an :py:class:`~.AlgebraicNumber`, specifically, either a :py:class:`~.ComplexRootOf`, or, optionally and where possible, an expression in radicals. Parameters ========== radicals : boolean, optional (default=True) If ``True``, then we will try to return the root as an expression in radicals. If that is not possible, we will return a :py:class:`~.ComplexRootOf`. minpoly : :py:class:`~.Poly` If the minimal polynomial for `self` has been pre-computed, it can be passed in order to save time. rrrN) rrLrr~rrrr same_root)rrrrrootsr?r]s r?rAlgebraicNumber.to_root s(  $ $Z ?-S-S99   0t..0 X . u:?8O \\^A{{1!!re)rrCr:rD)TN)!rrsrtrurvryr is_algebraicr;rrzsetrrErrDrrNrrrrrrrrrrrrrr|r}r~s@r?r~r~s"DILI D #uL*$fP"+&& :?K%?" )HV// *E'N!,rer~c"\rSrSrSrSrSrSrg)RationalConstanti z Abstract base class for rationals with specific behaviors Derived classes must define class attributes p and q and should probably all be singletons. rc.[R"U5$r:rrr s r?rRationalConstant.__new__ !!#&&reN)rrsrtrurvryrr|rrer?rr s I'rerc\rSrSrSrSrSrg)IntegerConstanti rc.[R"U5$r:rr s r?rIntegerConstant.__new__ rreN)rrsrtruryrr|rrer?rr s I'rercp\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr \S 5r\S 5rS rS rS rSrg)r'i zThe number zero. Zero is a singleton, and can be accessed by ``S.Zero`` Examples ======== >>> from sympy import S, Integer >>> Integer(0) is S.Zero True >>> 1/S.Zero zoo References ========== .. [1] https://en.wikipedia.org/wiki/Zero rrFTrcgrQrrs r?rmZero.__getnewargs__ rec"[R$r:rdrrer?r Zero.__abs__ vv rec"[R$r:rdrrer?r Zero.__neg__ rrecUR(aU$UR(a[R$URSLa[R $UR (a[R$UR5up#UR(a[RU-$U[RLaX-$gr) rrr rrIrr"rr`rS)rrnrtermss r?rZero._eval_power s  $ $K  $ $$$ $  E )55L <<55L ((*    $$e+ +  ;  recU$r:rrrs r?rZero._eval_order  recgrrrs r?r Zero.__bool__ sreN)rrsrtrurvrzr|r#rSr"r;r%ryrm staticmethodrrrrrr|rrer?r'r' sm& A AKKGIMI&rer') metaclasscv\rSrSrSrSrSrSrSrSr Sr \ S5r \ S5r S rS r\ S S j5rSrg )ri zThe number one. One is a singleton, and can be accessed by ``S.One``. Examples ======== >>> from sympy import S, Integer >>> Integer(1) is S.One True References ========== .. [1] https://en.wikipedia.org/wiki/1_%28number%29 TrrcgrQrrs r?rmOne.__getnewargs__0 rrec"[R$r:r^rrer?r One.__abs__3 uu rec"[R$r:)r rTrrer?r One.__neg__7 s }}recU$r:rrrns r?rOne._eval_power; rrecgr:rrs r?rOne._eval_order> sreNc4U(a[R$0$r:r^)rrrrrrs r?r One.factorsA s 55LIrer)rrsrtrurvr;r#rzr|ryrmr rrrrrr|rrer?rr ss IK A AICH&+rercT\rSrSrSrSrSrSrSrSr \ S5r \ S 5r S r Srg ) rTiJ a)The number negative one. NegativeOne is a singleton, and can be accessed by ``S.NegativeOne``. Examples ======== >>> from sympy import S, Integer >>> Integer(-1) is S.NegativeOne True See Also ======== One References ========== .. [1] https://en.wikipedia.org/wiki/%E2%88%921_%28number%29 TrxrrcgrQrrs r?rmNegativeOne.__getnewargs__h rrec"[R$r:r^rrer?rNegativeOne.__abs__k rrec"[R$r:r^rrer?rNegativeOne.__neg__o rrecUR(a[R$UR(a[R$[ U[ 5(Ga.[ U[5(a[S5U-$U[RLa[R$U[R[R4;a[R$U[RLa[R$[ U[5(a|URS:Xa&[R[UR 5-$[#UR UR5up#U(aX-U[X1R5--$g)Ngr5)is_oddr rTrrrLrrQrrrrfr rr|rrzr)rrnr>rs r?rNegativeOne._eval_powers s ;;== <<55L dF # #$&&T{D((quu}uu  A$6$677uu qvv~&$))66Q;??GDFFO;;dffdff-74!VV)<#<<<reN)rrsrtrurvr;rzr|ryrmr rrrr|rrer?rTrTJ sO,I A AIrerTc>\rSrSrSrSrSrSrSrSr \ S5r Sr g ) rfi zThe rational number 1/2. Half is a singleton, and can be accessed by ``S.Half``. Examples ======== >>> from sympy import S, Rational >>> Rational(1, 2) is S.Half True References ========== .. [1] https://en.wikipedia.org/wiki/One_half Trr5rcgrQrrs r?rmHalf.__getnewargs__ rrec"[R$r:)r rfrrer?r Half.__abs__ rreN) rrsrtrurvr;rzr|ryrmr rr|rrer?rfrf s6 I A AIrerfc^\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr SrSrSrSS jrSS jr\"S \5S 5r\r\"S \5S 5r\"S \5S5r\"S \5S5r\r\"S \5S5rSrSrSrSrU4Sjr Sr!Sr"\#RHr$\#RJr%\#RLr&\#RNr'\"S \5S5r(\(r)Sr*Sr+Sr,U=r-$)ri a[Positive infinite quantity. Explanation =========== In real analysis the symbol `\infty` denotes an unbounded limit: `x\to\infty` means that `x` grows without bound. Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled `+\infty` and `-\infty` can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. Infinity is a singleton, and can be accessed by ``S.Infinity``, or can be imported as ``oo``. Examples ======== >>> from sympy import oo, exp, limit, Symbol >>> 1 + oo oo >>> 42/oo 0 >>> x = Symbol('x') >>> limit(exp(x), x, oo) oo See Also ======== NegativeInfinity, NaN References ========== .. [1] https://en.wikipedia.org/wiki/Infinity TFrc.[R"U5$r:rr s r?rInfinity.__new__ rrecg)Nz\inftyrrprinters r?_latexInfinity._latex recX:XaU$gr:rrs r?r Infinity._eval_subs  ;J rec[S5$NrrQrs r?rInfinity._eval_evalf s U|rec $URU5$r:rrrkoptionss r?r&Infinity.evalf %%rerc[U[5(aK[R(a6U[R [R 4;a[R $U$[RX5$r:)rLrr4rr rrrrs r?rInfinity.__add__ L eV $ $):)C)C++QUU33uu K~~d**rec[U[5(aK[R(a6U[R [R 4;a[R $U$[RX5$r:)rLrr4rr rrrrs r?rInfinity.__sub__ J eV $ $):)C)CQUU++uu K~~d**rec&U*RU5$r:rrs r?r~Infinity.__rsub__ u%%rec.[U[5(al[R(aWUR(dU[ R La[ R $UR(aU$[ R$[RX5$r:) rLrr4rr"r rrrr$rs r?r$Infinity.__mul__ s[ eV $ $):)C)C}}uu )) %% %~~d**recX[U[5(a[R(alU[R Ld&U[R LdU[RLa[R$UR(aU$[R $[RX5$r: rLrr4rr rrris_extended_nonnegativer(rs r?r(Infinity.__truediv__ sq eV $ $):)C)C "+++QUUNuu ,, %% %!!$..rec"[R$r:r rrs r?rInfinity.__abs__ zzrec"[R$r:)r rrs r?rInfinity.__neg__ s!!!rec\UR(a[R$UR(a[R$U[R La[R $U[R La[R $URSLaUR(aSSK J n U"U5nUR(a[R $UR(a[R$UR(a[R $XR5-$gg)au ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``oo ** nan`` ``nan`` ``oo ** -p`` ``0`` ``p`` is number, ``oo`` ================ ======= ============================== See Also ======== Pow NaN NegativeInfinity Fr)r N)rr rrr'rrrIr;rr r#rSr"r&)rrnr  expt_reals r?rInfinity._eval_power s$  $ $::   $ $66M 155=55L 1$$ $55L  E )dnn ?4I$$((($$vv   uu % %/= )rec"[R$r:)rr+rs r?rInfinity._as_mpf_valF s yyrec >[TU]5$r:rBrEs r?rDInfinity.__hash__I rGrecPU[RL=(d U[S5:H$r6r rrrs r?r+Infinity.__eq__L s ";euU|&;;recPU[RL=(a U[S5:g$r6r]rs r?r/Infinity.__ne__O sAJJ&@5E%L+@@recX[U[5(d[$[R$r:rLrrr rrs r?rInfinity.__mod__W %&&! !uu recU$r:rrs r?rInfinity.floor_ rrecU$r:rrs r?rInfinity.ceilingb rrer:).rrsrtrurvrwr; is_complexrIrr%rrIryrr/r rr&rrrrrr~r$rr(rrrrrDr+r/rr;r?r4r8rrrrr|r}r~s@r?rr s^&PNIJKMHI'&(+)+ H(+)+(&)&(+)+H( /) /"$&L"<A[[F [[F [[F [[F() Hrerc^\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr SrSrSrSS jrSS jr\"S \5S 5r\r\"S \5S 5r\"S \5S5r\"S \5S5r\r\"S \5S5rSrSrSrSrU4Sjr Sr!Sr"\#RHr$\#RJr%\#RLr&\#RNr'\"S \5S5r(\(r)Sr*Sr+Sr,Sr-U=r.$)rih zNegative infinite quantity. NegativeInfinity is a singleton, and can be accessed by ``S.NegativeInfinity``. See Also ======== Infinity TFrc.[R"U5$r:rr s r?rNegativeInfinity.__new__ rrecg)Nz-\inftyrr-s r?r/NegativeInfinity._latex srecX:XaU$gr:rrs r?r NegativeInfinity._eval_subs r4rec[S5$Nrr7rs r?rNegativeInfinity._eval_evalf s V}rec $URU5$r:r:r;s r?r&NegativeInfinity.evalf r>rerc[U[5(aK[R(a6U[R [R 4;a[R $U$[RX5$r:)rLrr4rr rrrrs r?rNegativeInfinity.__add__ rDrec[U[5(aK[R(a6U[R [R 4;a[R $U$[RX5$r:)rLrr4rr rrrrs r?rNegativeInfinity.__sub__ rArec&U*RU5$r:rFrs r?r~NegativeInfinity.__rsub__ rHrec.[U[5(al[R(aWUR(dU[ R La[ R $UR(aU$[ R$[RX5$r:) rLrr4rr"r rrrr$rs r?r$NegativeInfinity.__mul__ sY eV $ $):)C)C}}uu )) :: ~~d**recX[U[5(a[R(alU[R Ld&U[R LdU[RLa[R$UR(aU$[R $[RX5$r:rLrs r?r(NegativeInfinity.__truediv__ so eV $ $):)C)C "+++QUUNuu ,, :: !!$..rec"[R$r:rPrs r?rNegativeInfinity.__abs__ rRrec"[R$r:rPrs r?rNegativeInfinity.__neg__ rRrecxUR(Ga(U[RLd&U[RLdU[RLa[R$[ U[ 5(aBUR(a1UR(a[R$[R$[RU-n[RU-nUS:XaUR(aU$U[RLa2UR(a!UR(d[R$X2-$g)a ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``(-oo) ** nan`` ``nan`` ``(-oo) ** oo`` ``nan`` ``(-oo) ** -oo`` ``nan`` ``(-oo) ** e`` ``oo`` ``e`` is positive even integer ``(-oo) ** o`` ``-oo`` ``o`` is positive odd integer ================ ======= ============================== See Also ======== Infinity Pow NaN rN) r;r rrrrLrrr"rTrrr")rrninf_parts_parts r?rNegativeInfinity._eval_power s, >>>quu} "A...uu $((T-F-F;;---::%zz4'H]]D(F1}!1!1A---$$V^^(((? "' rec"[R$r:)rr,rs r?rNegativeInfinity._as_mpf_val s zzrec >[TU]5$r:rBrEs r?rDNegativeInfinity.__hash__ rGrecPU[RL=(d U[S5:H$rrr rrrs r?r+NegativeInfinity.__eq__ s!***DeuV}.DDrecPU[RL=(a U[S5:g$rrrrs r?r/NegativeInfinity.__ne__ s!A...I5E&M3IIrecX[U[5(d[$[R$r:rbrs r?rNegativeInfinity.__mod__ rdrecU$r:rrs r?rNegativeInfinity.floor rrecU$r:rrs r?rNegativeInfinity.ceiling rrecF[RS[RS0$r")r rTrrs r?as_powers_dictNegativeInfinity.as_powers_dict s q!**a00rer:)/rrsrtrurvrIrirwrr%rr;rIryrr/r rr&rrrrrr~r$rr(rrrrrDr+r/rr;r?r4r8rrrrrr|r}r~s@r?rrh sc JNKMIHI'&(+)+ H(+)+(&)&(+)+H( /) /)#V"EJ[[F [[F [[F [[F() H11rercp^\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr SrSrSrSrSrSrSrSrS r\"S \5S 5r\"S \5S 5r\"S \5S 5r\"S \5S5rSrSrSrU4Sjr Sr!Sr"\#RHr$\#RJr%\#RLr&\#RNr'Sr(U=r)$)ri a Not a Number. Explanation =========== This serves as a place holder for numeric values that are indeterminate. Most operations on NaN, produce another NaN. Most indeterminate forms, such as ``0/0`` or ``oo - oo` produce NaN. Two exceptions are ``0**0`` and ``oo**0``, which all produce ``1`` (this is consistent with Python's float). NaN is loosely related to floating point nan, which is defined in the IEEE 754 floating point standard, and corresponds to the Python ``float('nan')``. Differences are noted below. NaN is mathematically not equal to anything else, even NaN itself. This explains the initially counter-intuitive results with ``Eq`` and ``==`` in the examples below. NaN is not comparable so inequalities raise a TypeError. This is in contrast with floating point nan where all inequalities are false. NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported as ``nan``. Examples ======== >>> from sympy import nan, S, oo, Eq >>> nan is S.NaN True >>> oo - oo nan >>> nan + 1 nan >>> Eq(nan, nan) # mathematical equality False >>> nan == nan # structural equality True References ========== .. [1] https://en.wikipedia.org/wiki/NaN TNFrc.[R"U5$r:rr s r?r NaN.__new__P rrecg)Nz \text{NaN}rr-s r?r/ NaN._latexS srecU$r:rrs r?r NaN.__neg__V rrercU$r:rrs r?r NaN.__add__Y rrecU$r:rrs r?r NaN.__sub__] rrecU$r:rrs r?r$ NaN.__mul__a rrecU$r:rrs r?r(NaN.__truediv__e rrecU$r:rrs r?r NaN.floori rrecU$r:rrs r?r NaN.ceilingl rrec[$r:)rrs r?rNaN._as_mpf_valo srec >[TU]5$r:rBrEs r?rD NaN.__hash__r rGrec&U[RL$r:r rrs r?r+ NaN.__eq__u s~rec&U[RL$r:rrs r?r/ NaN.__ne__y sAEE!!re)*rrsrtrurvrwrIrHrFris_transcendentalrr%rr"rIr#rSr;ryrr/rrrrrr$r(rrrrDr+r/rr;r?r4r8r|r}r~s@r?rr s.^NGKLJMIGHKKII'()()()()""[[F [[F [[F [[Frercgrr)r\r]s r? _eval_is_eqr  recv\rSrSrSrSrSrSrSrSr Sr \ r Sr SrSr\S5rS rS r\S 5rS rSrg )ri aComplex infinity. Explanation =========== In complex analysis the symbol `\tilde\infty`, called "complex infinity", represents a quantity with infinite magnitude, but undetermined complex phase. ComplexInfinity is a singleton, and can be accessed by ``S.ComplexInfinity``, or can be imported as ``zoo``. Examples ======== >>> from sympy import zoo >>> zoo + 42 zoo >>> 42/zoo 0 >>> zoo + zoo nan >>> zoo*zoo zoo See Also ======== Infinity TFrc.[R"U5$r:rr s r?rComplexInfinity.__new__ rrecg)Nz\tilde{\infty}rr-s r?r/ComplexInfinity._latex s rec"[R$r:rPrrer?rComplexInfinity.__abs__ s zzrecU$r:rrs r?rComplexInfinity.floor rrecU$r:rrs r?rComplexInfinity.ceiling rrec"[R$r:)r rrrer?rComplexInfinity.__neg__ s   recU[RLa[R$[U[5(aRUR (a[R$UR (a[R$[R$gr:)r rrrLrr"r#r'rs r?rComplexInfinity._eval_power sY 1$$ $55L dF # #||uu ##,,,66M $reN)rrsrtrurvrwrr;rIrirIrrzryrr/r rrrrrr|rrer?rr ss>NKIHJ DI'!!! "rercp^\rSrSrSrSrSrSrSr\ r Sr Sr Sr SrSrS rS rS rU4S jrSrU=r$) ri Trc.[R"U5$r:rr s r?rNumberSymbol.__new__ rrecg)z|Return an interval with number_cls endpoints that contains the value of NumberSymbol. If not implemented, then return None. Nrr number_clss r? approximationNumberSymbol.approximation srecL[RURU5U5$r:rrs r?rNumberSymbol._eval_evalf rrec[U5nXLagUR(aUR(agg![a [s$f=fr)r rrrxrGrs r?r+NumberSymbol.__eq__ sF "UOE = ??t11 "! ! "s 6A A c X:X+$r:rrs r?r/NumberSymbol.__ne__ rrecVXLa[R$[R"X5$r:)r truerr8rs r?r8NumberSymbol.__le__ =66M{{4''recVXLa[R$[R"X5$r:)r rrr?rs r?r?NumberSymbol.__ge__rrec[er:rrs r?rNumberSymbol.__int__ s!!rec >[TU]5$r:rBrEs r?rDNumberSymbol.__hash__rGre)rrsrtrurwrr;ryrrrzrrrr+r/r8r?rrDr|r}r~s@r?rr sSNIIIO D' 8 !( ( """rercx\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr \S5rSrS rS rS rS rS rSrSrg)Exp1iaThe `e` constant. Explanation =========== The transcendental number `e = 2.718281828\ldots` is the base of the natural logarithm and of the exponential function, `e = \exp(1)`. Sometimes called Euler's number or Napier's constant. Exp1 is a singleton, and can be accessed by ``S.Exp1``, or can be imported as ``E``. Examples ======== >>> from sympy import exp, log, E >>> E is exp(1) True >>> log(E) 1 References ========== .. [1] https://en.wikipedia.org/wiki/E_%28mathematical_constant%29 TFrcg)Nrrr-s r?r/ Exp1._latex8rMrec"[R$r:)r rrrer?r Exp1.__abs__;rrecgr,rrs r?r Exp1.__int__?rec[U5$r:)r(rs r?rExp1._as_mpf_valBs T{rec[U[5(a[S5[S54$[U[5(agg)Nr5r issubclassrrrs r?approximation_intervalExp1.approximation_intervalEs7 j' * *AJ + +  H - - .recj[R(aURU5$SSKJn U"U5$)Nr)r)r4 exp_is_pow_eval_power_exp_is_pow&sympy.functions.elementary.exponentialr)rrnrs r?rExp1._eval_powerKs)  ' '..t4 4 Bt9 recUR(a*U[La[$U[*:Xa[R$SSKJn [ X5(aURS$UR(Gd[[-nXU*4;a[$UR[[-5nU(GaSU-R(aUR(a[R$UR (a[R"$U[R$-R(a[*$U[R$-R (a[$O[UR&(aJUS-nUS:aUS-nXT:wa5[R(U[R*-[R,--$UR/5upFU[[*4;agU/Sp[0R2"U5HMn [ X5(aUcU RSnM' gU R4(aUR7U 5 MM g U(a U[1U6-$S$UR(a/n /n Sn URHn U [RLaU R7U 5 M)X -n[ U[85(aQUR:ULaBUR<U :waU R7UR<5 Sn MU R7U 5 MU R7U5 M U (dU (a[1U 6[9U[?U 6SS9-$gUR@(aUR=5$g)Nr)logr5rFTr)!rxoor r'rrrLris_AddIrrpirrrr"rTrfrRrPir r`Mul make_argsr%appendrbaserr is_Matrix)rrrIoorncoeffrrlog_termtermoutadd argchangedr\newas r?rExp1._eval_power_exp_is_powRs ==by vv > c  88A; B$CSDk! IIbdOEeG''}} uu  }},!&&.11 !r !&&.00 1&&"QYFz!  vvqtt AOO(CDD++-LEbS ! %wH e,d(('#'99Q<''MM$'-.68S&\) ?4 ? ZZCCJXX:JJqMwdC((TYY$->xx1} 488,%)  1 JJt$jCyT39u!EEE! ]]779 rec vSSKJn U"[[RS- -5[U"[5-- $)Nr)sinr5)(sympy.functions.elementary.trigonometricr rr r)rrVr s r?_eval_rewrite_as_sinExp1._eval_rewrite_as_sins(@1qttAv:3q6))rec vSSKJn U"[5[U"[[RS- -5--$)Nr)cosr5)r rrr r)rrVrs r?_eval_rewrite_as_cosExp1._eval_rewrite_as_coss)@1v#a!$$q&j/)))reN)rrsrtrurvrHr#rSrGr;rrryr/r rrrrrrrrr|rrer?rrsp6GKKMILI N`**rerc`\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr \S5rSrS rS rSrg ) riaThe `\pi` constant. Explanation =========== The transcendental number `\pi = 3.141592654\ldots` represents the ratio of a circle's circumference to its diameter, the area of the unit circle, the half-period of trigonometric functions, and many other things in mathematics. Pi is a singleton, and can be accessed by ``S.Pi``, or can be imported as ``pi``. Examples ======== >>> from sympy import S, pi, oo, sin, exp, integrate, Symbol >>> S.Pi pi >>> pi > 3 True >>> pi.is_irrational True >>> x = Symbol('x') >>> sin(x + 2*pi) sin(x) >>> integrate(exp(-x**2), (x, -oo, oo)) sqrt(pi) References ========== .. [1] https://en.wikipedia.org/wiki/Pi TFrcg)Nz\pirr-s r?r/ Pi._latexsrec"[R$r:)r rrrer?r Pi.__abs__s tt recg)Nrrrs r?r Pi.__int__rrec[U5$r:)r'rs r?rPi._as_mpf_vals d|rec[U[5(a[S5[S54$[U[5(a[SSS5[SSS54$g)NrrGrrrs r?rPi.approximation_intervalsQ j' * *AJ + +  H - -S"a((2q!*<= =.reN)rrsrtrurvrHr#rSrGr;rrryr/r rrrrr|rrer?rrs[!FGKKMILI>rercZ\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr SrSrS rS r\rSrg ) GoldenRatioia3The golden ratio, `\phi`. Explanation =========== `\phi = \frac{1 + \sqrt{5}}{2}` is an algebraic number. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities, i.e. their maximum. GoldenRatio is a singleton, and can be accessed by ``S.GoldenRatio``. Examples ======== >>> from sympy import S >>> S.GoldenRatio > 1 True >>> S.GoldenRatio.expand(func=True) 1/2 + sqrt(5)/2 >>> S.GoldenRatio.is_irrational True References ========== .. [1] https://en.wikipedia.org/wiki/Golden_ratio TFrcg)Nz\phirr-s r?r/GoldenRatio._latexsrecgr"rrs r?rGoldenRatio.__int__rrecf[R"[US-5U*S- 5n[X!5$NrJ)r from_man_expr)rqrs r?rGoldenRatio._as_mpf_vals.   y3dURZ @!!rec bSSKJn [R[RU"S5--$)Nr)sqrtrK)(sympy.functions.elementary.miscellaneousr.r rf)rhintsr.s r?_eval_expand_funcGoldenRatio._eval_expand_func#s AvvtAw&&rec[U[5(a[R[ S54$[U[5(aggr,rrr rrrs r?r"GoldenRatio.approximation_interval'7 j' * *EE8A;' '  H - - .reN)rrsrtrurvrHr#rSrGr;rrryr/rrr1r_eval_rewrite_as_sqrtr|rrer?r$r$sS8GKKMILI" ' .rer$c`\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr SrSrS rS rS r\rSrg ) TribonacciConstanti0aThe tribonacci constant. Explanation =========== The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial `x^3 - x^2 - x - 1 = 0`, and also satisfies the equation `x + x^{-3} = 2`. TribonacciConstant is a singleton, and can be accessed by ``S.TribonacciConstant``. Examples ======== >>> from sympy import S >>> S.TribonacciConstant > 1 True >>> S.TribonacciConstant.expand(func=True) 1/3 + (19 - 3*sqrt(33))**(1/3)/3 + (3*sqrt(33) + 19)**(1/3)/3 >>> S.TribonacciConstant.is_irrational True >>> S.TribonacciConstant.n(20) 1.8392867552141611326 References ========== .. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers TFrcg)Nz\text{TribonacciConstant}rr-s r?r/TribonacciConstant._latex_s+recgr"rrs r?rTribonacciConstant.__int__brrec8URU5R$r:)rrwrs r?rTribonacciConstant._as_mpf_vales%+++recVURSS9RUS-5n[X!S9$)NT)functionrr)r1rrQrs r?rTribonacciConstant._eval_evalfhs/  # #T # 2 > >tax HR((rec pSSKJnJn SU"SSU"S5-- 5-U"SSU"S5--5-S- $)Nr)cbrtr.rr!)r/rDr.)rr0rDr.s r?r1$TribonacciConstant._eval_expand_funcls<GDaRj))DaRj,AAQFFrec[U[5(a[R[ S54$[U[5(aggr,r4rs r?r)TribonacciConstant.approximation_intervalpr6reN)rrsrtrurvrHr#rSrGr;rrryr/rrrr1rr7r|rrer?r9r90sZ"HGKKMILI,,)G .rer9cH\rSrSrSrSrSrSrSrSr Sr Sr Sr S r S rSrg) EulerGammaiyaThe Euler-Mascheroni constant. Explanation =========== `\gamma = 0.5772157\ldots` (also called Euler's constant) is a mathematical constant recurring in analysis and number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm: .. math:: \gamma = \lim\limits_{n\to\infty} \left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right) EulerGamma is a singleton, and can be accessed by ``S.EulerGamma``. Examples ======== >>> from sympy import S >>> S.EulerGamma.is_irrational >>> S.EulerGamma > 0 True >>> S.EulerGamma > 1 False References ========== .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant TFNrcg)Nz\gammarr-s r?r/EulerGamma._latexr1recgrrrs r?rEulerGamma.__int__rrec[RRUS-5n[R"X!*S- 5n[ X15$r*)rlibhyper euler_fixedr+rqrrkrArps r?rEulerGamma._as_mpf_vals; MM % %dRi 0   q%"* -!!rec[U[5(a [R[R4$[U[ 5(a[R [ SSS54$g)NrrKr)rrr r'rrrfrs r?r!EulerGamma.approximation_intervalsK j' * *FFAEE? "  H - -FFHQ1-. ..re)rrsrtrurvrHr#rSrGr;ryr/rrrr|rrer?rKrKys;>GKKMII" /rerKcR\rSrSrSrSrSrSrSrSr Sr Sr Sr S r S S jrS rSrg) CatalaniaCatalan's constant. Explanation =========== $G = 0.91596559\ldots$ is given by the infinite series .. math:: G = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2} Catalan is a singleton, and can be accessed by ``S.Catalan``. Examples ======== >>> from sympy import S >>> S.Catalan.is_irrational >>> S.Catalan > 0 True >>> S.Catalan > 1 False References ========== .. [1] https://en.wikipedia.org/wiki/Catalan%27s_constant TFNrcgrrrs r?rCatalan.__int__rrec~[R"US-5n[R"X!*S- 5n[X15$r*)r catalan_fixedr+rqrSs r?rCatalan._as_mpf_vals7   tby )   q%"* -!!rec[U[5(a [R[R4$[U[ 5(a[ SSS5[R4$g)N rJr)rrr r'rrrs r?rCatalan.approximation_intervalsK j' * *FFAEE? "  H - -QA&. ..rec UcUbU$SSKJn SSKJn U"SSSS9nU"[R U-SU-S-S-- US[R 45$) Nrrr)Sumr@T)integer nonnegativer5)rrsympy.concrete.summationsrbr rTr)rk_symrr0rrbr@s r?_eval_rewrite_as_SumCatalan._eval_rewrite_as_SumsX  7#6K!1 #t 61==!#qs1uqj01a2DEErecg)NGrr-s r?r/Catalan._latexsrerC)rrsrtrurvrHr#rSrGr;ryrrrrgr/r|rrer?rXrXsA6GKKMII" / FrerXcv\rSrSrSrSrSrSrSrSr Sr \ r Sr Sr\S5rSrS rS rS r\S 5rSrg )r iaThe imaginary unit, `i = \sqrt{-1}`. I is a singleton, and can be accessed by ``S.I``, or can be imported as ``I``. Examples ======== >>> from sympy import I, sqrt >>> sqrt(-1) I >>> I*I -1 >>> 1/I -I References ========== .. [1] https://en.wikipedia.org/wiki/Imaginary_unit TFrc URS$)Nimaginary_unit_latex) _settingsr-s r?r/ImaginaryUnit._latexs  !788rec"[R$r:r^rrer?rImaginaryUnit.__abs__rrecU$r:rrs r?rImaginaryUnit._eval_evalfrrec$[R*$r:)r r rs r?rImaginaryUnit._eval_conjugatesrec[U[5(a^US-nUS:Xa[R$US:Xa[R$US:Xa[R $US:Xa[R*$[U[ 5(aL[US5up#[[RUSS9nUS-(a[[R USS9$U$g) z b is I = sqrt(-1) e is symbolic object but not equal to 0, 1 I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal I**0 mod 4 -> 1 I**1 mod 4 -> I I**2 mod 4 -> -1 I**3 mod 4 -> -I rrrr5rFrN) rLrr rr rTrrrr)rrnr>rrps r?rImaginaryUnit._eval_power"s dG $ $!8Dqyuu &}}$'' dH % %$?DAQ__a%8B1u1=="u==I &recB[R[R4$r:)r rTrfrs r? as_base_expImaginaryUnit.as_base_exp?s}}aff$$recV[S5R[S5R4$)Nrr)rQrwrs r?_mpc_ImaginaryUnit._mpc_Bsaa//reN)rrsrtrurvrw is_imaginaryrr;rrrrzryr/r rrrrrzrNr}r|rrer?r r ss,NLIIL DI9 :%00rer c[U[[45(ag[U5[LaUR 5$[U[ 5(ag[U[5(aURSS:$g)zreturn True only for a literal Number whose internal representation as a fraction has a denominator of 1, else False, i.e. integer, with no fractional part. Tr5rF) rLrrWrrrrrQrwr2s r?rrJsc !j#&'' Aw%||~!W!UwwqzQ rec[U5n[U5nUR(dUR(dX:H$UR(a*UR(aURUR:H$UR(aXpUR(dgURup#pEURUR pvU(aU*nUS:XaUS:H=(a X6:H$US:a4US:wagUR 5UR 5U-:wagX4-U:H$Xc:wagU*nUR 5S- U:wagSU-U:H$)a*Compare expressions treating plain floats as rationals. Examples ======== >>> from sympy import S, symbols, Rational, Float >>> from sympy.core.numbers import equal_valued >>> equal_valued(1, 2) False >>> equal_valued(1, 1) True In SymPy expressions with Floats compare unequal to corresponding expressions with rationals: >>> x = symbols('x') >>> x**2 == x**2.0 False However an individual Float compares equal to a Rational: >>> Rational(1, 2) == Float(0.5) False In a future version of SymPy this might change so that Rational and Float compare unequal. This function provides the behavior currently expected of ``==`` so that it could still be used if the behavior of ``==`` were to change in future. >>> equal_valued(1, 1.0) # Float vs Rational True >>> equal_valued(S(1).n(3), S(1).n(5)) # Floats of different precision True Explanation =========== In future SymPy versions Float and Rational might compare unequal and floats with different precisions might compare unequal. In that context a function is needed that can check if a number is equal to 1 or 0 etc. The idea is that instead of testing ``if x == 1:`` if we want to accept floats like ``1.0`` as well then the test can be written as ``if equal_valued(x, 1):`` or ``if equal_valued(x, 2):``. Since this function is intended to be used in situations where one or both operands are expected to be concrete numbers like 1 or 0 the function does not recurse through the args of any compound expression to compare any nested floats. References ========== .. [1] https://github.com/sympy/sympy/pull/20033 Frr)r r$rwrRrzr| bit_length) r2r rlrmrr`rzr|neg_exps r?rr[sj  A A ::ajjv  ww!''!! 1 == Ds 33q d axAv"#(" q 6 <<>S^^-3 3zQ 8$ <<>A  (W ""recv^^^^^^^[[4mUU4SjmUUUU4SjmUU4SjmUUU4SjmT"X5$)ayReturn True if expr1 and expr2 are numerically close. The expressions must have the same structure, but any Rational, Integer, or Float numbers they contain are compared approximately using rtol and atol. Any other parts of expressions are compared exactly. However, allowance is made to allow for the additive and multiplicative identities. Relative tolerance is measured with respect to expr2 so when used in testing expr2 should be the expected correct answer. Examples ======== >>> from sympy import exp >>> from sympy.abc import x, y >>> from sympy.core.numbers import all_close >>> expr1 = 0.1*exp(x - y) >>> expr2 = exp(x - y)/10 >>> expr1 0.1*exp(x - y) >>> expr2 exp(x - y)/10 >>> expr1 == expr2 False >>> all_close(expr1, expr2) True Identities are automatically supplied: >>> all_close(x, x + 1e-10) True >>> all_close(x, 1.0*x) True >>> all_close(x, 1.0*x + 1e-10) True c>[U5[U5:XaR[U[[45(a7[ U5[ U5:wag[ U4Sj[ X555$T"[U5[U55$)NFc38># UHupT"X5v M g7fr:r)r=e1e2 _all_closes r?r@0all_close.._all_close..sHfbz")))rrLrrrrOrUr )obj1obj2r_all_close_exprs r?rall_close.._all_closes` :d # 4$(G(G4yCI%HDHH H"8D>8D>B Brec>[UT5n[UT5nX#:wagU(aT"X5$UR(d3UR(d"UR(dUR(aT"X5$URUR:wd,[ UR 5[ UR 5:wag[ UR UR 5n[U4SjU55$)NFc38># UHupT"X5v M g7fr:r)r=a1a2rs r?r@5all_close.._all_close_expr..s>vr?2**r)rLris_MulrrrrUrO) expr1expr2num1num2r NUM_TYPES _all_close_acr _close_nums r?r"all_close.._all_close_exprs%+%+ < e+ + <<5<<5<<5<< . . :: #s5::#ejj/'I5::uzz*>>>>recT>[[X- 5TT[U5--:*5$r:)rrX)rratolrtols r?rall_close.._close_nums'C $tCI~(==>>rec>UR(dUR(GaURSS9up#URSS9upET"X$5(dg[U5n[U5nXg-nXh-nXx-nU(dg[SXg455(dg[ U5V s/sHoR 5PM nn [ U5V s/sHoR 5PM nn [ [[U555n UH1n U H(n Xyn T"X5(dMU RU 5 M/ g U (+$UR(dUR(deUR5n UR5nT"U SUS5(dg[U 5[U5:wag[ U 5H8nX;dM T"U RU5URU55(aM8 g U (dgU H4nUH+nT"UU5(dMT"U UUU5(d g M2 g gs sn fs sn f)NFrNTc3\# UH"oHo"R[5v M M$ g7fr:)hasrQ)r=rIr>s r?r@3all_close.._all_close_ac..s AXq!uuU||q|Xs*,r) rrWranyrrzrrRrremoveras_coefficients_dictpop)rrc1rc2rs1s2commonr> unmatchedbe1be2cd1cd2r@k1k2rrrs r?r all_close.._all_close_acs  <<5<<<'''7FB'''7FBb%%RBRBWF LB LBAbXAAA,32;7;a--/;B7+22;7;a--/;B7U3r7^,I"A%C!#++!((+ # !!> !||u||++((*((*#a&#a&)) s8s3x cAx!#''!*cggaj99 B"2r** &c"gs2w77$[87s %H? I)rrQ) rrrrrrrrrs ``@@@@@r? all_closers9L5!IC ? ??BH e ##recgrrrs r?rrDrrecX[RURURS5$r")rrr[r\)rs r?sympify_fractionsrIs ==ammQ 77rec*[[U55$r:rrWrs r? sympify_mpzrQs1vrecf[[UR5[UR55$r:rrWr[r\rs r? sympify_mpqrT!AKK(#amm*<==rec*[[U55$r:rrs r? sympify_fmpzr]rrecf[[UR5[UR55$r:rrs r? sympify_fmpqr`rrecV[R"XRR5$r:)r _from_mpmathrrkrs r?sympify_mpmathrgs   Q  //rec[[[URUR455upU[ R U--$r:)rmapr realimagr r )r\rrs r?sympify_complexrms6c'AFFAFF#345JD !//$& &&re)r)r)rc[RR[5 [RR[5 [R R[ 5 [R R[5 gr:)rVrregisterRealrQrrWrrrer?_register_classesrzsN NNF# LL%  h' g&rer:rr)gh㈵>g:0yE>) __future__rtypingrrVrrYr containersrr rrr r r singletonr rbasicrrrrr&rcacherr decoratorsrintfuncrrrrrlogicrrzrsortingrsympy.external.gmpyrrr sympy.multipledispatchr!ry mpmath.libmplibmprr"r#rirgr%r&r'r(r)mpmath.ctx_mp_pythonr*mpmath.libmp.libmpfr+rr,rr-rr.r/rhr0r1sympy.utilities.miscr2sympy.utilities.exceptionsr3 parametersr4r_LOG2rHrqrtrur}rrMrJrKrLrrrrQrr RealNumberrrrWr~rrr'rrTrfrrrrrrrzoorrErrr$r9rKrXr rrrrrrZrrrmpzmpqrrfmpzfmpqrrcomplexpowerrmulridentityrrr_illegalrrer?rs" ,,#"'"II77+ 7$::*'@)  jZ< e  $  }}T]]<013tv&Zv&r _ ?F_ ?F?DC*7??; lvl^G hG T EdEP 'x ''g'A?iAH0/Y0f=/Y=@ y@yvyvZZd1vd1Nq&Iqfee #tJ"jIJ"Z6":6"rV*<9V*pFF?>?>BTT<.,)<.~F.F.R8/8/v<li<~Q0J)Q0hOO"^#BE$P %8(9##$>+6T$((1+&'-8T$((1a.)* >-9T%**Q-()/;T%**Q*+,0-',u v '  EE1::q1113D3D Ere